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Research ArticleAnalytical Solutions for Systems of Singular PartialDifferential-Algebraic Equations
U Filobello-Nino1 H Vazquez-Leal1 B Benhammouda2 A Perez-Sesma1
V M Jimenez-Fernandez1 J Cervantes-Perez1 A Sarmiento-Reyes3
J Huerta-Chua4 L J Morales-Mendoza5 M Gonzalez-Lee5 A Diaz-Sanchez3
D Pereyra-Diacuteaz1 and R Loacutepez-Martiacutenez6
1Facultad de Instrumentacion Electronica Universidad Veracruzana Circuito Gonzalo Aguirre Beltran SN91000 Xalapa VER Mexico2Higher Colleges of Technology Abu Dhabi Menrsquos College PO Box 25035 Abu Dhabi UAE3Instituto Nacional de Astrofısica Optica y Electronica Luis Enrique Erro 1 72840 Santa Marıa Tonantzintla PUE Mexico4Facultad de Ingenierıa Civil Universidad Veracruzana Venustiano Carranza SN Colonia Revolucion93390 Poza Rica VER Mexico5Departamento de Ingenierıa Electronica Universidad Veracruzana Venustiano Carranza SNColonia Revolucion 93390 Poza Rica VER Mexico6Facultad de Matematicas Universidad Veracruzana Circuito Gonzalo Aguirre Beltran SN 91000 Xalapa VER Mexico
Correspondence should be addressed to H Vazquez-Leal hvazquezuvmx
Received 13 October 2014 Accepted 17 December 2014
Academic Editor Baodong Zheng
Copyright copy 2015 U Filobello-Nino et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper proposes power series method (PSM) in order to find solutions for singular partial differential-algebraic equations(SPDAEs) We will solve three examples to show that PSMmethod can be used to search for analytical solutions of SPDAEs Whatis more we will see that in some cases Pade posttreatment besides enlarging the domain of convergence may be employed inorder to get the exact solution from the truncated series solutions of PSM
1 Introduction
The importance of research on partial differential-algebraicequations (PDAEs) is that they are used in the mathematicalmodeling of many phenomena both practical and theoret-ical These systems arise for example in nanoelectronicselectrical networks and mechanical systems among manyothers Despite the importance of this topic it may beconsidered relatively new and little known
Although the case of constant-coefficient linear PDAEshas been investigated by means of numerical methods forinstance in [1 2] perhaps themore relevant aspect of PDAEsboth linear and nonlinear is the concept of index Thedifferentiation index is defined as the minimum number oftimes that all or part of the PDAEs must be differentiatedwith respect to time in order to obtain the time derivative of
the solution as a continuous function of the solution and itsspace derivatives [3] A fact that justifies the search for othermethods of solution to these equations is that the solutionsof higher index PDAEs (index greater than one) becomevery complicated even for numerical methods and manyapplication problems lead to PDAEs with different indicesA further difficulty to be considered that arises and affectsalso other kinds of systems of differential equations as well asdifferential equations is the presence of singularities whichare related to points at which some terms of the differentialequations become infinite or undefined
In recent years several methods focused on approxi-mating nonlinear and linear problems as an alternative toclassical methods have been reported such as those based onvariational approaches [4ndash7] tanh method [8] exp-function[9 10] Adomianrsquos decomposition method [11ndash16] parameter
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 752523 9 pageshttpdxdoiorg1011552015752523
2 Discrete Dynamics in Nature and Society
expansion [17] homotopy perturbation method [7 16 18ndash46] homotopy analysis method [47] homotopy asymptoticmethod [48] series method [49 50] and perturbationmethod [51ndash54] among many others Also a few exact solu-tions to nonlinear differential equations have been reportedoccasionally [55]
This study shows that power seriesmethod (PSM) [56 57]is able to address the above difficulties to obtain power seriessolutions for singular partial differential-algebraic equations(SPDAEs) that is PDAEswith singular pointsThese systemsturn out to be difficult even for numerical methods Moregenerally we will see that the combination of PSM andPade posttreatment could be effective to improve the PSMrsquostruncated series solutions in convergence rate what is moresometimes it ends up giving the exact solution of the systemsuch as what will happen in our third case study
This paper is organized as follows In Section 2 weintroduce the basic idea of power series method Section 3provides a brief explanation of application of PSM to solveSPDAEs Section 4 presents three case studies one singu-lar nonlinear index-one system one singular linear index-two system and one singular nonlinear index-two systemBesides a discussion on the results is presented in Section 5Finally a brief conclusion is given in Section 6
2 Basic Concept of Power Series Method
It can be considered that a nonlinear differential equation canbe expressed as
119860 (119906) minus 119891 (119905) = 0 119905 isin Ω (1)
with the following boundary condition
119861(119906120597119906
120597119899) = 0 119905 isin Γ (2)
where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119905) is a known analytical function and Γ is thedomain boundary forΩ
PSM [49 50] assumes that the solution of a differentialequation can be written in the following form
119906 (119905) =
infin
sum
119899=0
119906119899119905119899
(3)
where 1199060 1199061 are unknown functions to be determined by
series methodThe method of solution for differential equations can be
summarized as follows
(1) Equation (3) is substituted into (1) and then weregroup the resulting polynomial equation in termsof powers of 119905
(2) We equate each coefficient of the above-mentionedpolynomial to zero
(3) As a consequence a linear algebraic system for theunknowns of (3) is obtained
(4) To conclude the solution of the above system allowsobtaining the coefficients 119906
0 1199061
3 Application of PSM to Solve PDAE Systems
Sincemany applications problems in science and engineeringare often modeled by semiexplicit PDAEs we considertherefore the following class of PDAEs
1199061119905= 120601 (119906 119906
119909 119906119909119909) (4)
0 = 120595 (119906 119906119909 119906119909119909) (119905 119909) isin (0 119879) times (119886 119887) (5)
where 119906119896 [0 119879]times [119886 119887] rarr 119877
119898119896 119896 = 1 2 and 119887 gt 119886 in otherwords 119906 = (119906
1 1199062)
For clarification the method is described for the generalsystem (4)-(5) where the number of unknowns is givenby 1198981+ 1198982 In this notation 119906
1(differential unknown)
has 1198981components and 119906
2(algebraic unknown) has 119898
2
components In fact 1198981and 119898
2can take any values greater
than or equal to one so that the number of unknowns in (4)-(5) is greater than or equal to 2
System (4)-(5) is subject to the initial condition
1199061(0 119909) = 119892 (119909) 119886 le 119909 le 119887 (6)
and some suitable boundary conditions
119861 (119906 (119905 119886) 119906 (119905 119887) 119906119909(119905 119886) 119906
119909(119905 119887)) = 0 0 le 119905 le 119879 (7)
where 119892(119909) is a given functionWe assume that the solution to initial boundary-value
problem (4)ndash(7) exists and is unique and sufficiently smoothTo simplify the exposition of the PSM we integrate first
(4) with respect to 119905 and use the initial condition (6) to obtain
1199061(119905 119909) minus 119892 (119909) minus int
119905
0
120601 (119906 119906119909 119906119909119909) 119889119905 = 0 (8)
It is important to note that the time integration of (4) is notrelevant to the solution procedure presented here so one canapply the PSM directly to (4)
A fact that justifies the use of PSM is that in general termsgetting solutions for PDAEs becomes very complicated evenfor numerical methods Moreover there are not systematicanalytical or numerical methods to solve these problems
In view of PSM we assume the solution components119906119896(119905 119909) 119896 = 1 2 have the form
119906119896(119905 119909) = 119906
1198960(119909) + 119906
1198961(119909) 119905 + 119906
1198962(119909) 1199052
+ sdot sdot sdot (9)
where 119906119896119899(119909) 119896 = 1 2 119899 = 0 1 2 are unknown functions
to be determined later on by the PSMThen substitute (9) into system (4)-(5) and equate the
coefficients of powers of 119905 in the resulting equation to zero toobtain an algebraic linear system for the coefficients whosesolution is employed in (9) with the end of obtaining asolution for (4)ndash(7) in series form These series may havelimited regions of convergence even if we take a large numberof terms Therefore in some cases it will be convenient toapply the Pade resummationmethod to PSM truncated seriesto enlarge the convergence region as depicted in the nextsectionA relevant fact is that the steps outlined in this sectionwill be also sufficient to obtain satisfactory solutions for themost difficult case of SPDAEs
Discrete Dynamics in Nature and Society 3
4 Case Studies
The objective of this section is employing PSM in order tosolve three SPDAE systems
Our results will show the efficiency of the presentedmethod
41 Nonlinear Index-One SPDAE (following Section 31198981= 1
and1198982= 1) Consider the following
1199061119905minus 1199061119909119909
+ 11990611199061119909+1199062
119909= 1199093
minus 6119909119905 + 31199052
1199095
+1199054
1199092 (10)
1199061+ 1199062= 1199051199093
+1199054
119909 119905 gt 0 (11)
subject to the initial conditions
1199061(0 119909) = 0 (12)
In order to apply PSM we integrate (10) with respect to 119905 anduse the initial condition (12) to obtain1199061(119905 119909)
= int
119905
0
[1199061119909119909
minus 11990611199061119909minus1199062
119909+ 1199093
minus 6119909119905 + 31199052
1199095
+1199054
1199092]119889119905
(13)
PSM assumes that 119906(119905 119909) and V(119905 119909) can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (14)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (15)
where 11990610(119909) 11990611(119909) 11990612(119909) 11990620(119909) 11990621(119909) 11990622(119909) are un-
known functionsThis case study is simplified substituting (14) and (15) into
(11) to getinfin
sum
119899=0
1199062119899119905119899
=1199054
119909+ 1199051199093
minus
infin
sum
119899=0
1199061119899119905119899
(16)
On the other hand substituting (14) through (16) into (13)leads to
infin
sum
119899=0
1199061119899119905119899
= int
119905
0
[
infin
sum
119899=0
11990610158401015840
1119899119905119899
minus
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898119905119899+119898
minus1
119909(1199054
119909+ 1199051199093
minus
infin
sum
119899=0
1199061119899119905119899
)
+1199093
minus 6119909119905 + 31199052
1199095
+1199054
1199092]119889119905
(17)
From here on the dash notation in 1199061015840 denotes the ordinaryderivative with respect to 119909
Integrating the above result it is obtained thatinfin
sum
119899=0
1199061119899119905119899
=
infin
sum
119899=0
11990610158401015840
1119899
119905119899+1
119899 + 1minus
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898
119905119899+119898+1
119899 + 119898 + 1
minus1199092
1199052
2+ 1199093
119905 minus 31199091199052
+ 1199095
1199053
+1
119909
infin
sum
119899=0
1199061119899119905119899+1
119899 + 1
(18)
Standardizing the summation index and grouping we get therecursive formula
119906101199050
minus 1199093
119905 + (3119909 +1199092
2) 1199052
minus 1199095
1199053
+
infin
sum
119896=1
[1199061119896minus11990610158401015840
1119896minus1
119896+
infin
sum
119898=0
1199061119896minus119898minus1
1199061015840
1119898
119896minus1199061119896minus1
119896119909] 119905119896
= 0
(19)
Equating the coefficients of powers of 119905 to zero in (19) weobtain
119896 = 0
1199060= 0
119896 = 1
11990611= 11990610158401015840
10minus 1199061015840
1011990610+1
11990911990610+ 1199093
(20)
after employing (20) it is obtained that
11990611= 1199093
119896 = 2
11990612=11990610158401015840
11
2minus1199061015840
1011990611
2minus1199061015840
1111990610
2+11990611
2119909minus1199092
2minus 3119909
(21)
substituting (20) and (21) in the above equation it is obtainedthat
11990612= 0
119896 = 3
11990613=11990610158401015840
12
3minus1199061015840
1011990612
3minus1199061015840
1111990611
3minus1199061015840
1211990610
3+11990612
3119909+ 1199095
(22)
after substituting (20) (21) and (22) in the last equation weget
11990613= 0
119896 = 4
11990614=11990610158401015840
13
4minus1199061015840
1011990613
4minus1199061015840
1111990612
4minus1199061015840
1211990611
4minus1199061015840
1311990610
4+11990613
4119909
(23)
after employing (20) (21) (22) and (23) we get
11990614= 0
119896 = 5
11990615=11990610158401015840
14
5minus1199061015840
1011990614
5minus1199061015840
1111990613
5minus1199061015840
1211990612
5
minus1199061015840
1311990611
5minus1199061015840
1411990610
5+11990614
5119909
(24)
4 Discrete Dynamics in Nature and Society
the substitution of (20) (21) (22) (23) and (24) leads to
11990615= 0 (25)
in the same way we obtain
11990616= 11990617= 11990618= sdot sdot sdot = 0 (26)
Substituting (20) through (26) into (14) leads us to
1199061(119905 119909) = 119909
3
119905 (27)
Finally substituting (27) into (11) leads to
1199062(119905 119909) =
1199054
119909 (28)
Thus (27) and (28) are the exact solution for SPDAE system(10)ndash(12)
42 Linear Index-Two SPDAEwithVariable Coefficients (1198981=
2 1198982= 1) Consider the following
1199061119905= 1199092
1199061119909119909
minus 31199061+ 1199063+
1199092
1 + 119905 (29)
1199062119905= 1199092
1199062119909119909
minus 31199062+ 1199063+
1199092
1 + 119905 (30)
0 = 1199061+ 1199062minus 21199092 ln (1 + 119905) (31)
subject to the initial conditions
1199061(0 119909) = 0 119906
2(0 119909) = 0
minus1 lt 119905 le 1 minusinfin lt 119909 lt infin
(32)
The integration of (29) and (30) with respect to 119905 and usingthe initial conditions (32) lead to
1199061(119905 119909) = int
119905
0
[1199092
1199061119909119909
minus 31199061+ 1199063] 119889119905 + 119909
2 ln (1 + 119905) (33)
1199062(119905 119909) = int
119905
0
[1199092
1199062119909119909
minus 31199062+ 1199063] 119889119905 + 119909
2 ln (1 + 119905) (34)
assuming that 1199061(119905 119909) 119906
2(119905 119909) and 119906
3(119905 119909) can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (35)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (36)
1199063(119905 119909) = 119906
30(119909) + 119906
31(119909) 119905 + 119906
32(119909) 1199052
+ sdot sdot sdot (37)
where 11990610(119909) 11990611(119909) 119906
20(119909) 11990621(119909) 119906
30(119909) 11990631(119909)
are unknown functions to be determined later on by thePSM method
After substituting (35) and (37) into (33) we get
119906101199050
+
infin
sum
119896=1
1
119896[1198961199061119896minus 1199092
11990610158401015840
1119896minus1+ 31199061119896minus1
minus1199063119896minus1
minus 1199092
(minus1)119896minus1
] 119905119896
= 0
(38)
where we have standardized the summation index and em-ployed the following Taylor series expansion
ln (1 + 119905) =infin
sum
119899=1
(minus1)119899minus1
119899119905119899
minus1 lt 119905 le 1 (39)
In the same way the substitution of (36) and (37) into (34)leads to
119906201199050
+
infin
sum
119896=1
1
119896[1198961199062119896minus 1199092
11990610158401015840
2119896minus1+ 31199062119896minus1
minus1199063119896minus1
minus 1199092
(minus1)119896minus1
] 119905119896
= 0
(40)
On the other hand after substituting (35) (36) and (39) into(31) we have
infin
sum
119896=1
[1199061119896+ 1199062119896minus21199092
119896(minus1)119896minus1
] 119905119896
= 0 (41)
where we have employed the following results deduced from(38) and (40)
11990610= 11990620= 0 (42)
Equations (38) (40) and (41) give rise to the followingformulas
1199061119899=1199092
11990610158401015840
1119899minus1minus 31199061119899minus1
+ 1199063119899minus1
+ (minus1)119899minus1
1199092
119899 119899 ge 1 (43)
1199062119899=1199092
11990610158401015840
2119899minus1minus 31199062119899minus1
+ 1199063119899minus1
+ (minus1)119899minus1
1199092
119899 119899 ge 1 (44)
1199061119899+ 1199062119899=21199092
(minus1)119899minus1
119899 119899 ge 1 (45)
Combining the result of adding (43) and (44) with (45) weobtain
1199063119899minus1
= minus1
2(11990610158401015840
1119899minus1+ 11990610158401015840
2119899minus1) 1199092
+3
2(1199061119899minus1
+ 1199062119899minus1
) 119899 ge 1
(46)
The substitution of (46) into (43) and (44) respectively leadsus to
1199061119899=
1
2119899(1199092
11990610158401015840
1119899minus1minus 31199061119899minus1
+ 31199062119899minus1
minus1199092
11990610158401015840
2119899minus1+ 2 (minus1)
119899minus1
1199092
) 119899 ge 1
1199062119899=
1
2119899(1199092
11990610158401015840
2119899minus1minus 31199062119899minus1
+ 31199061119899minus1
minus1199092
11990610158401015840
1119899minus1+ 2 (minus1)
119899minus1
1199092
) 119899 ge 1
(47)
Discrete Dynamics in Nature and Society 5
From recursion formulas (46) and (47) we get the functions
11990610(119909) = 0 119906
11(119909) = 119909
2
11990612(119909) =
minus1199092
2
11990613=1199092
3 119906
14=minus1199092
4sdot sdot sdot
(48)
11990620(119909) = 0 119906
21(119909) = 119909
2
11990622(119909) =
minus1199092
2
11990623=1199092
3 119906
24=minus1199092
4sdot sdot sdot
(49)
11990630(119909) = 0 119906
31(119909) = 119909
2
11990632(119909) =
minus1199092
2
11990633=1199092
3 119906
34=minus1199092
4sdot sdot sdot
(50)
After substituting (48) through (50) into series (35) (36) and(37) respectively we get
1199061(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (51)
1199062(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (52)
1199063(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (53)
After identifying the 119899th terms of the series (51) (52) and (53)as ((minus1)119899minus1119899)119905119899 we conclude that
1199061(119905 119909) = 119909
2 ln (1 + 119905)
1199062(119905 119909) = 119909
2 ln (1 + 119905)
1199063(119905 119909) = 119909
2 ln (1 + 119905)
(54)
which is the exact solution of (29)ndash(32) (see (39))
43 Nonlinear Index-Two SPDAE with Variable Coefficients(1198981= 2 119898
2= 1) Finally consider the following
1199061119905= 119891 (119909) 119906
1119909119909+ 11990611199061119909minus1 minus 119905
1 + 1199051199063 (55)
1199062119905= 119892 (119909) 119906
2119909119909minus 11990621199062119909+1 + 119905
1 minus 1199051199063 (56)
0 = 1199061(1 + 119905) minus 119906
2(1 minus 119905) minusinfin lt 119909 lt infin minus1 lt 119905 lt 1
(57)
subject to the initial conditions
1199061(0 119909) = 119909 119906
2(0 119909) = 119909 119906
3(0 119909) = 2119909 (58)
where 119891(119909) and 119892(119909) are analytical functions on minusinfin lt 119909 lt
infin
The integration of (55) and (56)with respect to 119905 andusingthe initial conditions (58) lead to
1199061(119905 119909) = 119909 + int
119905
0
[119891 (119909) 1199061119909119909
+ 11990611199061119909minus1 minus 119905
1 + 1199051199063] 119889119905 (59)
1199062(119905 119909) = 119909 + int
119905
0
[119892 (119909) 1199062119909119909
minus 11990621199062119909+1 + 119905
1 minus 1199051199063] 119889119905 (60)
PSM assumes once again that 1199061(119905 119909) 119906
2(119905 119909) and 119906
3(119905 119909)
can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (61)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (62)
1199063(119905 119909) = 119906
30(119909) + 119906
31(119909) 119905 + 119906
32(119909) 1199052
+ sdot sdot sdot (63)
where 11990610(119909) 11990611(119909) 119906
20(119909) 11990621(119909) 119906
30(119909) 11990631(119909)
are unknown functions to be determined later on by thePSM method
Substituting (61) and (63) into (59) and also (62) and (63)into (60) respectively we getinfin
sum
119899=0
1199061119899119905119899
= 119909 + int
119905
0
119891 (119909)
infin
sum
119899=0
11990610158401015840
1119899119905119899
119889119905 + int
119905
0
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898119905119899+119898
119889119905
minus int
119905
0
(1 minus 119905)
infin
sum
119899=0
infin
sum
119898=0
(minus1)119899
1199063119898119905119899+119898
119889119905
(64)infin
sum
119899=0
1199062119899119905119899
= 119909 + int
119905
0
119892 (119909)
infin
sum
119899=0
11990610158401015840
2119899119905119899
119889119905 + int
119905
0
infin
sum
119899=0
infin
sum
119898=0
11990621198991199061015840
2119898119905119899+119898
119889119905
minus int
119905
0
(1 + 119905)
infin
sum
119899=0
infin
sum
119898=0
1199063119898119905119899+119898
119889119905
(65)
where we have employed the Taylor series expansions
1
1 minus 119905=
infin
sum
119899=0
119905119899
1
1 + 119905=
infin
sum
119899=0
(minus1)119899
119905119899
(66)
After integrating and standardizing the summation indexwe get the following recursion formulas from (64) and (65)respectively
minus 11990610+ 119909 minus 119906
30119905 minus
1
2(11990631minus 211990630) 1199052
minus1
3(11990632minus 211990631+ 211990630) 1199053
minus1
4(11990633minus 211990632+ 211990631minus 211990630) 1199054
+
infin
sum
119896=1
[119891 (119909) 119906
10158401015840
1119896minus1
119896+
infin
sum
119898=0
1199061015840
11198981199061119896minus119898minus1
119896minus 1199061119896] 119905119896
= 0
minus 11990620+ 119909 + 119906
30119905 +
1
2(11990631+ 211990630) 1199052
+1
3(11990632+ 211990631+ 211990630) 1199053
6 Discrete Dynamics in Nature and Society
+1
4(11990633+ 211990632+ 211990631+ 211990630) 1199054
+
infin
sum
119896=1
[119892 (119909) 119906
10158401015840
2119896minus1
119896minus
infin
sum
119898=0
1199061015840
21198981199062119896minus119898minus1
119896minus 1199062119896] 119905119896
= 0
(67)
From (57) we obtain
infin
sum
119898=0
1199062119898119905119898
= (1 + 119905)
infin
sum
119899=0
infin
sum
119895=0
119905119899+119895
1199061119895 (68)
after using again the first series of (66)After standardizing the summation index we get a third
recurrence formula from (68)
1199062119896=
infin
sum
119899=0
[1199061119896minus119899
+ 1199061119896minus119899minus1
] where 119896 = 0 1 2 3
(69)
From recursion formulas (67) and (69) we get the followingcoupled equations
11990610= 11990610(0 119909) (70)
11990611= 119891 (119909) 119906
10158401015840
10+ 1199061015840
1011990610minus 11990630 (71)
11990612= 119891 (119909)
11990610158401015840
11
2+1199061015840
1011990611+ 1199061015840
1111990610
2+11990631
2 (72)
11990613= 119891 (119909)
11990610158401015840
12
3+1199061015840
1011990612+ 1199061015840
1111990611+ 1199061015840
1211990610
3
minus11990632+ 211990630minus 211990631
3
(73)
11990614=11990610158401015840
13
4+1199061015840
1011990613+ 1199061015840
1111990612+ 1199061015840
1211990611+ 1199061015840
1311990610
4
minus11990633minus 211990632+ 211990631minus 211990630
4
(74)
11990620= 11990620(0 119909) (75)
11990621= 119892 (119909) 119906
10158401015840
20minus 1199061015840
2011990620+ 11990630 (76)
11990622= 119892 (119909)
11990610158401015840
21
2minus1199061015840
2011990621+ 1199061015840
2111990620
2+11990631+ 211990630
2 (77)
11990623= 119892 (119909)
11990610158401015840
22
3minus1199061015840
2011990622+ 1199061015840
2111990621+ 1199061015840
2211990620
3
+11990632+ 211990630+ 211990631
3
(78)
11990624= 119892 (119909)
11990610158401015840
23
4minus1199061015840
2011990623+ 1199061015840
2111990622+ 1199061015840
2211990621+ 1199061015840
2311990620
4
+11990633+ 211990632+ 211990631+ 211990630
4
(79)
11990620= 11990610 (80)
11990621= 211990610+ 11990611 (81)
11990622= 11990612+ 211990611+ 211990610 (82)
11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)
11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610
(84)
From (70) through (84) we get the functions
11990610= 119909 119906
11= minus119909 119906
12= 119909
11990613= minus119909 119906
14= 119909 sdot sdot sdot
(85)
11990620= 119909 119906
21= 119909 119906
22= 119909
11990623= 119909 119906
24= 119909 sdot sdot sdot
(86)
11990630= 2119909 119906
31= 0 119906
32= 2119909
11990633= 0 119906
34= 2119909
(87)
Substituting (85) through (87) into series (61) (62) and (63)respectively we get
1199061(119905 119909) = 119909 (1 minus 119905 + 119905
2
minus 1199053
+ 1199054
+ sdot sdot sdot ) (88)
1199062(119905 119909) = 119909 (1 + 119905 + 119905
2
+ 1199053
+ 1199054
+ sdot sdot sdot ) (89)
1199063(119905 119909) = 2119909 (1 + 119905
2
+ 1199054
+ 1199056
+ sdot sdot sdot ) (90)
After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms
1199061(119905 119909) =
119909
1 + 119905
1199062(119905 119909) =
119909
1 minus 119905
1199063(119905 119909) =
2119909
1 minus 1199052
(91)
which is the exact solution of (55)ndash(58) where we haveemployed (66) and
1
1 minus 1199052=
infin
sum
119899=0
1199052119899
(92)
This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general
Discrete Dynamics in Nature and Society 7
terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)
5 Discussion
In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution
Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs
One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs
Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems
6 Conclusion
By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above
all of index greater than one combining PSM and Laplace-Pade resummation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024
References
[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998
[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999
[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000
[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007
[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007
[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007
[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[11] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional
8 Discrete Dynamics in Nature and Society
decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011
[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011
[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006
[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008
[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009
[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005
[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009
[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008
[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011
[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012
[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012
[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013
[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014
[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012
[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012
[42] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011
[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011
[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra
Discrete Dynamics in Nature and Society 9
and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012
[46] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011
[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956
[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906
[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995
[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995
[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013
[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013
[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014
[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011
[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Discrete Dynamics in Nature and Society
expansion [17] homotopy perturbation method [7 16 18ndash46] homotopy analysis method [47] homotopy asymptoticmethod [48] series method [49 50] and perturbationmethod [51ndash54] among many others Also a few exact solu-tions to nonlinear differential equations have been reportedoccasionally [55]
This study shows that power seriesmethod (PSM) [56 57]is able to address the above difficulties to obtain power seriessolutions for singular partial differential-algebraic equations(SPDAEs) that is PDAEswith singular pointsThese systemsturn out to be difficult even for numerical methods Moregenerally we will see that the combination of PSM andPade posttreatment could be effective to improve the PSMrsquostruncated series solutions in convergence rate what is moresometimes it ends up giving the exact solution of the systemsuch as what will happen in our third case study
This paper is organized as follows In Section 2 weintroduce the basic idea of power series method Section 3provides a brief explanation of application of PSM to solveSPDAEs Section 4 presents three case studies one singu-lar nonlinear index-one system one singular linear index-two system and one singular nonlinear index-two systemBesides a discussion on the results is presented in Section 5Finally a brief conclusion is given in Section 6
2 Basic Concept of Power Series Method
It can be considered that a nonlinear differential equation canbe expressed as
119860 (119906) minus 119891 (119905) = 0 119905 isin Ω (1)
with the following boundary condition
119861(119906120597119906
120597119899) = 0 119905 isin Γ (2)
where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119905) is a known analytical function and Γ is thedomain boundary forΩ
PSM [49 50] assumes that the solution of a differentialequation can be written in the following form
119906 (119905) =
infin
sum
119899=0
119906119899119905119899
(3)
where 1199060 1199061 are unknown functions to be determined by
series methodThe method of solution for differential equations can be
summarized as follows
(1) Equation (3) is substituted into (1) and then weregroup the resulting polynomial equation in termsof powers of 119905
(2) We equate each coefficient of the above-mentionedpolynomial to zero
(3) As a consequence a linear algebraic system for theunknowns of (3) is obtained
(4) To conclude the solution of the above system allowsobtaining the coefficients 119906
0 1199061
3 Application of PSM to Solve PDAE Systems
Sincemany applications problems in science and engineeringare often modeled by semiexplicit PDAEs we considertherefore the following class of PDAEs
1199061119905= 120601 (119906 119906
119909 119906119909119909) (4)
0 = 120595 (119906 119906119909 119906119909119909) (119905 119909) isin (0 119879) times (119886 119887) (5)
where 119906119896 [0 119879]times [119886 119887] rarr 119877
119898119896 119896 = 1 2 and 119887 gt 119886 in otherwords 119906 = (119906
1 1199062)
For clarification the method is described for the generalsystem (4)-(5) where the number of unknowns is givenby 1198981+ 1198982 In this notation 119906
1(differential unknown)
has 1198981components and 119906
2(algebraic unknown) has 119898
2
components In fact 1198981and 119898
2can take any values greater
than or equal to one so that the number of unknowns in (4)-(5) is greater than or equal to 2
System (4)-(5) is subject to the initial condition
1199061(0 119909) = 119892 (119909) 119886 le 119909 le 119887 (6)
and some suitable boundary conditions
119861 (119906 (119905 119886) 119906 (119905 119887) 119906119909(119905 119886) 119906
119909(119905 119887)) = 0 0 le 119905 le 119879 (7)
where 119892(119909) is a given functionWe assume that the solution to initial boundary-value
problem (4)ndash(7) exists and is unique and sufficiently smoothTo simplify the exposition of the PSM we integrate first
(4) with respect to 119905 and use the initial condition (6) to obtain
1199061(119905 119909) minus 119892 (119909) minus int
119905
0
120601 (119906 119906119909 119906119909119909) 119889119905 = 0 (8)
It is important to note that the time integration of (4) is notrelevant to the solution procedure presented here so one canapply the PSM directly to (4)
A fact that justifies the use of PSM is that in general termsgetting solutions for PDAEs becomes very complicated evenfor numerical methods Moreover there are not systematicanalytical or numerical methods to solve these problems
In view of PSM we assume the solution components119906119896(119905 119909) 119896 = 1 2 have the form
119906119896(119905 119909) = 119906
1198960(119909) + 119906
1198961(119909) 119905 + 119906
1198962(119909) 1199052
+ sdot sdot sdot (9)
where 119906119896119899(119909) 119896 = 1 2 119899 = 0 1 2 are unknown functions
to be determined later on by the PSMThen substitute (9) into system (4)-(5) and equate the
coefficients of powers of 119905 in the resulting equation to zero toobtain an algebraic linear system for the coefficients whosesolution is employed in (9) with the end of obtaining asolution for (4)ndash(7) in series form These series may havelimited regions of convergence even if we take a large numberof terms Therefore in some cases it will be convenient toapply the Pade resummationmethod to PSM truncated seriesto enlarge the convergence region as depicted in the nextsectionA relevant fact is that the steps outlined in this sectionwill be also sufficient to obtain satisfactory solutions for themost difficult case of SPDAEs
Discrete Dynamics in Nature and Society 3
4 Case Studies
The objective of this section is employing PSM in order tosolve three SPDAE systems
Our results will show the efficiency of the presentedmethod
41 Nonlinear Index-One SPDAE (following Section 31198981= 1
and1198982= 1) Consider the following
1199061119905minus 1199061119909119909
+ 11990611199061119909+1199062
119909= 1199093
minus 6119909119905 + 31199052
1199095
+1199054
1199092 (10)
1199061+ 1199062= 1199051199093
+1199054
119909 119905 gt 0 (11)
subject to the initial conditions
1199061(0 119909) = 0 (12)
In order to apply PSM we integrate (10) with respect to 119905 anduse the initial condition (12) to obtain1199061(119905 119909)
= int
119905
0
[1199061119909119909
minus 11990611199061119909minus1199062
119909+ 1199093
minus 6119909119905 + 31199052
1199095
+1199054
1199092]119889119905
(13)
PSM assumes that 119906(119905 119909) and V(119905 119909) can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (14)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (15)
where 11990610(119909) 11990611(119909) 11990612(119909) 11990620(119909) 11990621(119909) 11990622(119909) are un-
known functionsThis case study is simplified substituting (14) and (15) into
(11) to getinfin
sum
119899=0
1199062119899119905119899
=1199054
119909+ 1199051199093
minus
infin
sum
119899=0
1199061119899119905119899
(16)
On the other hand substituting (14) through (16) into (13)leads to
infin
sum
119899=0
1199061119899119905119899
= int
119905
0
[
infin
sum
119899=0
11990610158401015840
1119899119905119899
minus
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898119905119899+119898
minus1
119909(1199054
119909+ 1199051199093
minus
infin
sum
119899=0
1199061119899119905119899
)
+1199093
minus 6119909119905 + 31199052
1199095
+1199054
1199092]119889119905
(17)
From here on the dash notation in 1199061015840 denotes the ordinaryderivative with respect to 119909
Integrating the above result it is obtained thatinfin
sum
119899=0
1199061119899119905119899
=
infin
sum
119899=0
11990610158401015840
1119899
119905119899+1
119899 + 1minus
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898
119905119899+119898+1
119899 + 119898 + 1
minus1199092
1199052
2+ 1199093
119905 minus 31199091199052
+ 1199095
1199053
+1
119909
infin
sum
119899=0
1199061119899119905119899+1
119899 + 1
(18)
Standardizing the summation index and grouping we get therecursive formula
119906101199050
minus 1199093
119905 + (3119909 +1199092
2) 1199052
minus 1199095
1199053
+
infin
sum
119896=1
[1199061119896minus11990610158401015840
1119896minus1
119896+
infin
sum
119898=0
1199061119896minus119898minus1
1199061015840
1119898
119896minus1199061119896minus1
119896119909] 119905119896
= 0
(19)
Equating the coefficients of powers of 119905 to zero in (19) weobtain
119896 = 0
1199060= 0
119896 = 1
11990611= 11990610158401015840
10minus 1199061015840
1011990610+1
11990911990610+ 1199093
(20)
after employing (20) it is obtained that
11990611= 1199093
119896 = 2
11990612=11990610158401015840
11
2minus1199061015840
1011990611
2minus1199061015840
1111990610
2+11990611
2119909minus1199092
2minus 3119909
(21)
substituting (20) and (21) in the above equation it is obtainedthat
11990612= 0
119896 = 3
11990613=11990610158401015840
12
3minus1199061015840
1011990612
3minus1199061015840
1111990611
3minus1199061015840
1211990610
3+11990612
3119909+ 1199095
(22)
after substituting (20) (21) and (22) in the last equation weget
11990613= 0
119896 = 4
11990614=11990610158401015840
13
4minus1199061015840
1011990613
4minus1199061015840
1111990612
4minus1199061015840
1211990611
4minus1199061015840
1311990610
4+11990613
4119909
(23)
after employing (20) (21) (22) and (23) we get
11990614= 0
119896 = 5
11990615=11990610158401015840
14
5minus1199061015840
1011990614
5minus1199061015840
1111990613
5minus1199061015840
1211990612
5
minus1199061015840
1311990611
5minus1199061015840
1411990610
5+11990614
5119909
(24)
4 Discrete Dynamics in Nature and Society
the substitution of (20) (21) (22) (23) and (24) leads to
11990615= 0 (25)
in the same way we obtain
11990616= 11990617= 11990618= sdot sdot sdot = 0 (26)
Substituting (20) through (26) into (14) leads us to
1199061(119905 119909) = 119909
3
119905 (27)
Finally substituting (27) into (11) leads to
1199062(119905 119909) =
1199054
119909 (28)
Thus (27) and (28) are the exact solution for SPDAE system(10)ndash(12)
42 Linear Index-Two SPDAEwithVariable Coefficients (1198981=
2 1198982= 1) Consider the following
1199061119905= 1199092
1199061119909119909
minus 31199061+ 1199063+
1199092
1 + 119905 (29)
1199062119905= 1199092
1199062119909119909
minus 31199062+ 1199063+
1199092
1 + 119905 (30)
0 = 1199061+ 1199062minus 21199092 ln (1 + 119905) (31)
subject to the initial conditions
1199061(0 119909) = 0 119906
2(0 119909) = 0
minus1 lt 119905 le 1 minusinfin lt 119909 lt infin
(32)
The integration of (29) and (30) with respect to 119905 and usingthe initial conditions (32) lead to
1199061(119905 119909) = int
119905
0
[1199092
1199061119909119909
minus 31199061+ 1199063] 119889119905 + 119909
2 ln (1 + 119905) (33)
1199062(119905 119909) = int
119905
0
[1199092
1199062119909119909
minus 31199062+ 1199063] 119889119905 + 119909
2 ln (1 + 119905) (34)
assuming that 1199061(119905 119909) 119906
2(119905 119909) and 119906
3(119905 119909) can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (35)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (36)
1199063(119905 119909) = 119906
30(119909) + 119906
31(119909) 119905 + 119906
32(119909) 1199052
+ sdot sdot sdot (37)
where 11990610(119909) 11990611(119909) 119906
20(119909) 11990621(119909) 119906
30(119909) 11990631(119909)
are unknown functions to be determined later on by thePSM method
After substituting (35) and (37) into (33) we get
119906101199050
+
infin
sum
119896=1
1
119896[1198961199061119896minus 1199092
11990610158401015840
1119896minus1+ 31199061119896minus1
minus1199063119896minus1
minus 1199092
(minus1)119896minus1
] 119905119896
= 0
(38)
where we have standardized the summation index and em-ployed the following Taylor series expansion
ln (1 + 119905) =infin
sum
119899=1
(minus1)119899minus1
119899119905119899
minus1 lt 119905 le 1 (39)
In the same way the substitution of (36) and (37) into (34)leads to
119906201199050
+
infin
sum
119896=1
1
119896[1198961199062119896minus 1199092
11990610158401015840
2119896minus1+ 31199062119896minus1
minus1199063119896minus1
minus 1199092
(minus1)119896minus1
] 119905119896
= 0
(40)
On the other hand after substituting (35) (36) and (39) into(31) we have
infin
sum
119896=1
[1199061119896+ 1199062119896minus21199092
119896(minus1)119896minus1
] 119905119896
= 0 (41)
where we have employed the following results deduced from(38) and (40)
11990610= 11990620= 0 (42)
Equations (38) (40) and (41) give rise to the followingformulas
1199061119899=1199092
11990610158401015840
1119899minus1minus 31199061119899minus1
+ 1199063119899minus1
+ (minus1)119899minus1
1199092
119899 119899 ge 1 (43)
1199062119899=1199092
11990610158401015840
2119899minus1minus 31199062119899minus1
+ 1199063119899minus1
+ (minus1)119899minus1
1199092
119899 119899 ge 1 (44)
1199061119899+ 1199062119899=21199092
(minus1)119899minus1
119899 119899 ge 1 (45)
Combining the result of adding (43) and (44) with (45) weobtain
1199063119899minus1
= minus1
2(11990610158401015840
1119899minus1+ 11990610158401015840
2119899minus1) 1199092
+3
2(1199061119899minus1
+ 1199062119899minus1
) 119899 ge 1
(46)
The substitution of (46) into (43) and (44) respectively leadsus to
1199061119899=
1
2119899(1199092
11990610158401015840
1119899minus1minus 31199061119899minus1
+ 31199062119899minus1
minus1199092
11990610158401015840
2119899minus1+ 2 (minus1)
119899minus1
1199092
) 119899 ge 1
1199062119899=
1
2119899(1199092
11990610158401015840
2119899minus1minus 31199062119899minus1
+ 31199061119899minus1
minus1199092
11990610158401015840
1119899minus1+ 2 (minus1)
119899minus1
1199092
) 119899 ge 1
(47)
Discrete Dynamics in Nature and Society 5
From recursion formulas (46) and (47) we get the functions
11990610(119909) = 0 119906
11(119909) = 119909
2
11990612(119909) =
minus1199092
2
11990613=1199092
3 119906
14=minus1199092
4sdot sdot sdot
(48)
11990620(119909) = 0 119906
21(119909) = 119909
2
11990622(119909) =
minus1199092
2
11990623=1199092
3 119906
24=minus1199092
4sdot sdot sdot
(49)
11990630(119909) = 0 119906
31(119909) = 119909
2
11990632(119909) =
minus1199092
2
11990633=1199092
3 119906
34=minus1199092
4sdot sdot sdot
(50)
After substituting (48) through (50) into series (35) (36) and(37) respectively we get
1199061(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (51)
1199062(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (52)
1199063(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (53)
After identifying the 119899th terms of the series (51) (52) and (53)as ((minus1)119899minus1119899)119905119899 we conclude that
1199061(119905 119909) = 119909
2 ln (1 + 119905)
1199062(119905 119909) = 119909
2 ln (1 + 119905)
1199063(119905 119909) = 119909
2 ln (1 + 119905)
(54)
which is the exact solution of (29)ndash(32) (see (39))
43 Nonlinear Index-Two SPDAE with Variable Coefficients(1198981= 2 119898
2= 1) Finally consider the following
1199061119905= 119891 (119909) 119906
1119909119909+ 11990611199061119909minus1 minus 119905
1 + 1199051199063 (55)
1199062119905= 119892 (119909) 119906
2119909119909minus 11990621199062119909+1 + 119905
1 minus 1199051199063 (56)
0 = 1199061(1 + 119905) minus 119906
2(1 minus 119905) minusinfin lt 119909 lt infin minus1 lt 119905 lt 1
(57)
subject to the initial conditions
1199061(0 119909) = 119909 119906
2(0 119909) = 119909 119906
3(0 119909) = 2119909 (58)
where 119891(119909) and 119892(119909) are analytical functions on minusinfin lt 119909 lt
infin
The integration of (55) and (56)with respect to 119905 andusingthe initial conditions (58) lead to
1199061(119905 119909) = 119909 + int
119905
0
[119891 (119909) 1199061119909119909
+ 11990611199061119909minus1 minus 119905
1 + 1199051199063] 119889119905 (59)
1199062(119905 119909) = 119909 + int
119905
0
[119892 (119909) 1199062119909119909
minus 11990621199062119909+1 + 119905
1 minus 1199051199063] 119889119905 (60)
PSM assumes once again that 1199061(119905 119909) 119906
2(119905 119909) and 119906
3(119905 119909)
can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (61)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (62)
1199063(119905 119909) = 119906
30(119909) + 119906
31(119909) 119905 + 119906
32(119909) 1199052
+ sdot sdot sdot (63)
where 11990610(119909) 11990611(119909) 119906
20(119909) 11990621(119909) 119906
30(119909) 11990631(119909)
are unknown functions to be determined later on by thePSM method
Substituting (61) and (63) into (59) and also (62) and (63)into (60) respectively we getinfin
sum
119899=0
1199061119899119905119899
= 119909 + int
119905
0
119891 (119909)
infin
sum
119899=0
11990610158401015840
1119899119905119899
119889119905 + int
119905
0
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898119905119899+119898
119889119905
minus int
119905
0
(1 minus 119905)
infin
sum
119899=0
infin
sum
119898=0
(minus1)119899
1199063119898119905119899+119898
119889119905
(64)infin
sum
119899=0
1199062119899119905119899
= 119909 + int
119905
0
119892 (119909)
infin
sum
119899=0
11990610158401015840
2119899119905119899
119889119905 + int
119905
0
infin
sum
119899=0
infin
sum
119898=0
11990621198991199061015840
2119898119905119899+119898
119889119905
minus int
119905
0
(1 + 119905)
infin
sum
119899=0
infin
sum
119898=0
1199063119898119905119899+119898
119889119905
(65)
where we have employed the Taylor series expansions
1
1 minus 119905=
infin
sum
119899=0
119905119899
1
1 + 119905=
infin
sum
119899=0
(minus1)119899
119905119899
(66)
After integrating and standardizing the summation indexwe get the following recursion formulas from (64) and (65)respectively
minus 11990610+ 119909 minus 119906
30119905 minus
1
2(11990631minus 211990630) 1199052
minus1
3(11990632minus 211990631+ 211990630) 1199053
minus1
4(11990633minus 211990632+ 211990631minus 211990630) 1199054
+
infin
sum
119896=1
[119891 (119909) 119906
10158401015840
1119896minus1
119896+
infin
sum
119898=0
1199061015840
11198981199061119896minus119898minus1
119896minus 1199061119896] 119905119896
= 0
minus 11990620+ 119909 + 119906
30119905 +
1
2(11990631+ 211990630) 1199052
+1
3(11990632+ 211990631+ 211990630) 1199053
6 Discrete Dynamics in Nature and Society
+1
4(11990633+ 211990632+ 211990631+ 211990630) 1199054
+
infin
sum
119896=1
[119892 (119909) 119906
10158401015840
2119896minus1
119896minus
infin
sum
119898=0
1199061015840
21198981199062119896minus119898minus1
119896minus 1199062119896] 119905119896
= 0
(67)
From (57) we obtain
infin
sum
119898=0
1199062119898119905119898
= (1 + 119905)
infin
sum
119899=0
infin
sum
119895=0
119905119899+119895
1199061119895 (68)
after using again the first series of (66)After standardizing the summation index we get a third
recurrence formula from (68)
1199062119896=
infin
sum
119899=0
[1199061119896minus119899
+ 1199061119896minus119899minus1
] where 119896 = 0 1 2 3
(69)
From recursion formulas (67) and (69) we get the followingcoupled equations
11990610= 11990610(0 119909) (70)
11990611= 119891 (119909) 119906
10158401015840
10+ 1199061015840
1011990610minus 11990630 (71)
11990612= 119891 (119909)
11990610158401015840
11
2+1199061015840
1011990611+ 1199061015840
1111990610
2+11990631
2 (72)
11990613= 119891 (119909)
11990610158401015840
12
3+1199061015840
1011990612+ 1199061015840
1111990611+ 1199061015840
1211990610
3
minus11990632+ 211990630minus 211990631
3
(73)
11990614=11990610158401015840
13
4+1199061015840
1011990613+ 1199061015840
1111990612+ 1199061015840
1211990611+ 1199061015840
1311990610
4
minus11990633minus 211990632+ 211990631minus 211990630
4
(74)
11990620= 11990620(0 119909) (75)
11990621= 119892 (119909) 119906
10158401015840
20minus 1199061015840
2011990620+ 11990630 (76)
11990622= 119892 (119909)
11990610158401015840
21
2minus1199061015840
2011990621+ 1199061015840
2111990620
2+11990631+ 211990630
2 (77)
11990623= 119892 (119909)
11990610158401015840
22
3minus1199061015840
2011990622+ 1199061015840
2111990621+ 1199061015840
2211990620
3
+11990632+ 211990630+ 211990631
3
(78)
11990624= 119892 (119909)
11990610158401015840
23
4minus1199061015840
2011990623+ 1199061015840
2111990622+ 1199061015840
2211990621+ 1199061015840
2311990620
4
+11990633+ 211990632+ 211990631+ 211990630
4
(79)
11990620= 11990610 (80)
11990621= 211990610+ 11990611 (81)
11990622= 11990612+ 211990611+ 211990610 (82)
11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)
11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610
(84)
From (70) through (84) we get the functions
11990610= 119909 119906
11= minus119909 119906
12= 119909
11990613= minus119909 119906
14= 119909 sdot sdot sdot
(85)
11990620= 119909 119906
21= 119909 119906
22= 119909
11990623= 119909 119906
24= 119909 sdot sdot sdot
(86)
11990630= 2119909 119906
31= 0 119906
32= 2119909
11990633= 0 119906
34= 2119909
(87)
Substituting (85) through (87) into series (61) (62) and (63)respectively we get
1199061(119905 119909) = 119909 (1 minus 119905 + 119905
2
minus 1199053
+ 1199054
+ sdot sdot sdot ) (88)
1199062(119905 119909) = 119909 (1 + 119905 + 119905
2
+ 1199053
+ 1199054
+ sdot sdot sdot ) (89)
1199063(119905 119909) = 2119909 (1 + 119905
2
+ 1199054
+ 1199056
+ sdot sdot sdot ) (90)
After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms
1199061(119905 119909) =
119909
1 + 119905
1199062(119905 119909) =
119909
1 minus 119905
1199063(119905 119909) =
2119909
1 minus 1199052
(91)
which is the exact solution of (55)ndash(58) where we haveemployed (66) and
1
1 minus 1199052=
infin
sum
119899=0
1199052119899
(92)
This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general
Discrete Dynamics in Nature and Society 7
terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)
5 Discussion
In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution
Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs
One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs
Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems
6 Conclusion
By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above
all of index greater than one combining PSM and Laplace-Pade resummation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024
References
[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998
[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999
[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000
[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007
[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007
[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007
[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[11] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional
8 Discrete Dynamics in Nature and Society
decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011
[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011
[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006
[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008
[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009
[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005
[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009
[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008
[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011
[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012
[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012
[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013
[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014
[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012
[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012
[42] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011
[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011
[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra
Discrete Dynamics in Nature and Society 9
and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012
[46] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011
[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956
[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906
[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995
[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995
[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013
[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013
[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014
[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011
[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 3
4 Case Studies
The objective of this section is employing PSM in order tosolve three SPDAE systems
Our results will show the efficiency of the presentedmethod
41 Nonlinear Index-One SPDAE (following Section 31198981= 1
and1198982= 1) Consider the following
1199061119905minus 1199061119909119909
+ 11990611199061119909+1199062
119909= 1199093
minus 6119909119905 + 31199052
1199095
+1199054
1199092 (10)
1199061+ 1199062= 1199051199093
+1199054
119909 119905 gt 0 (11)
subject to the initial conditions
1199061(0 119909) = 0 (12)
In order to apply PSM we integrate (10) with respect to 119905 anduse the initial condition (12) to obtain1199061(119905 119909)
= int
119905
0
[1199061119909119909
minus 11990611199061119909minus1199062
119909+ 1199093
minus 6119909119905 + 31199052
1199095
+1199054
1199092]119889119905
(13)
PSM assumes that 119906(119905 119909) and V(119905 119909) can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (14)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (15)
where 11990610(119909) 11990611(119909) 11990612(119909) 11990620(119909) 11990621(119909) 11990622(119909) are un-
known functionsThis case study is simplified substituting (14) and (15) into
(11) to getinfin
sum
119899=0
1199062119899119905119899
=1199054
119909+ 1199051199093
minus
infin
sum
119899=0
1199061119899119905119899
(16)
On the other hand substituting (14) through (16) into (13)leads to
infin
sum
119899=0
1199061119899119905119899
= int
119905
0
[
infin
sum
119899=0
11990610158401015840
1119899119905119899
minus
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898119905119899+119898
minus1
119909(1199054
119909+ 1199051199093
minus
infin
sum
119899=0
1199061119899119905119899
)
+1199093
minus 6119909119905 + 31199052
1199095
+1199054
1199092]119889119905
(17)
From here on the dash notation in 1199061015840 denotes the ordinaryderivative with respect to 119909
Integrating the above result it is obtained thatinfin
sum
119899=0
1199061119899119905119899
=
infin
sum
119899=0
11990610158401015840
1119899
119905119899+1
119899 + 1minus
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898
119905119899+119898+1
119899 + 119898 + 1
minus1199092
1199052
2+ 1199093
119905 minus 31199091199052
+ 1199095
1199053
+1
119909
infin
sum
119899=0
1199061119899119905119899+1
119899 + 1
(18)
Standardizing the summation index and grouping we get therecursive formula
119906101199050
minus 1199093
119905 + (3119909 +1199092
2) 1199052
minus 1199095
1199053
+
infin
sum
119896=1
[1199061119896minus11990610158401015840
1119896minus1
119896+
infin
sum
119898=0
1199061119896minus119898minus1
1199061015840
1119898
119896minus1199061119896minus1
119896119909] 119905119896
= 0
(19)
Equating the coefficients of powers of 119905 to zero in (19) weobtain
119896 = 0
1199060= 0
119896 = 1
11990611= 11990610158401015840
10minus 1199061015840
1011990610+1
11990911990610+ 1199093
(20)
after employing (20) it is obtained that
11990611= 1199093
119896 = 2
11990612=11990610158401015840
11
2minus1199061015840
1011990611
2minus1199061015840
1111990610
2+11990611
2119909minus1199092
2minus 3119909
(21)
substituting (20) and (21) in the above equation it is obtainedthat
11990612= 0
119896 = 3
11990613=11990610158401015840
12
3minus1199061015840
1011990612
3minus1199061015840
1111990611
3minus1199061015840
1211990610
3+11990612
3119909+ 1199095
(22)
after substituting (20) (21) and (22) in the last equation weget
11990613= 0
119896 = 4
11990614=11990610158401015840
13
4minus1199061015840
1011990613
4minus1199061015840
1111990612
4minus1199061015840
1211990611
4minus1199061015840
1311990610
4+11990613
4119909
(23)
after employing (20) (21) (22) and (23) we get
11990614= 0
119896 = 5
11990615=11990610158401015840
14
5minus1199061015840
1011990614
5minus1199061015840
1111990613
5minus1199061015840
1211990612
5
minus1199061015840
1311990611
5minus1199061015840
1411990610
5+11990614
5119909
(24)
4 Discrete Dynamics in Nature and Society
the substitution of (20) (21) (22) (23) and (24) leads to
11990615= 0 (25)
in the same way we obtain
11990616= 11990617= 11990618= sdot sdot sdot = 0 (26)
Substituting (20) through (26) into (14) leads us to
1199061(119905 119909) = 119909
3
119905 (27)
Finally substituting (27) into (11) leads to
1199062(119905 119909) =
1199054
119909 (28)
Thus (27) and (28) are the exact solution for SPDAE system(10)ndash(12)
42 Linear Index-Two SPDAEwithVariable Coefficients (1198981=
2 1198982= 1) Consider the following
1199061119905= 1199092
1199061119909119909
minus 31199061+ 1199063+
1199092
1 + 119905 (29)
1199062119905= 1199092
1199062119909119909
minus 31199062+ 1199063+
1199092
1 + 119905 (30)
0 = 1199061+ 1199062minus 21199092 ln (1 + 119905) (31)
subject to the initial conditions
1199061(0 119909) = 0 119906
2(0 119909) = 0
minus1 lt 119905 le 1 minusinfin lt 119909 lt infin
(32)
The integration of (29) and (30) with respect to 119905 and usingthe initial conditions (32) lead to
1199061(119905 119909) = int
119905
0
[1199092
1199061119909119909
minus 31199061+ 1199063] 119889119905 + 119909
2 ln (1 + 119905) (33)
1199062(119905 119909) = int
119905
0
[1199092
1199062119909119909
minus 31199062+ 1199063] 119889119905 + 119909
2 ln (1 + 119905) (34)
assuming that 1199061(119905 119909) 119906
2(119905 119909) and 119906
3(119905 119909) can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (35)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (36)
1199063(119905 119909) = 119906
30(119909) + 119906
31(119909) 119905 + 119906
32(119909) 1199052
+ sdot sdot sdot (37)
where 11990610(119909) 11990611(119909) 119906
20(119909) 11990621(119909) 119906
30(119909) 11990631(119909)
are unknown functions to be determined later on by thePSM method
After substituting (35) and (37) into (33) we get
119906101199050
+
infin
sum
119896=1
1
119896[1198961199061119896minus 1199092
11990610158401015840
1119896minus1+ 31199061119896minus1
minus1199063119896minus1
minus 1199092
(minus1)119896minus1
] 119905119896
= 0
(38)
where we have standardized the summation index and em-ployed the following Taylor series expansion
ln (1 + 119905) =infin
sum
119899=1
(minus1)119899minus1
119899119905119899
minus1 lt 119905 le 1 (39)
In the same way the substitution of (36) and (37) into (34)leads to
119906201199050
+
infin
sum
119896=1
1
119896[1198961199062119896minus 1199092
11990610158401015840
2119896minus1+ 31199062119896minus1
minus1199063119896minus1
minus 1199092
(minus1)119896minus1
] 119905119896
= 0
(40)
On the other hand after substituting (35) (36) and (39) into(31) we have
infin
sum
119896=1
[1199061119896+ 1199062119896minus21199092
119896(minus1)119896minus1
] 119905119896
= 0 (41)
where we have employed the following results deduced from(38) and (40)
11990610= 11990620= 0 (42)
Equations (38) (40) and (41) give rise to the followingformulas
1199061119899=1199092
11990610158401015840
1119899minus1minus 31199061119899minus1
+ 1199063119899minus1
+ (minus1)119899minus1
1199092
119899 119899 ge 1 (43)
1199062119899=1199092
11990610158401015840
2119899minus1minus 31199062119899minus1
+ 1199063119899minus1
+ (minus1)119899minus1
1199092
119899 119899 ge 1 (44)
1199061119899+ 1199062119899=21199092
(minus1)119899minus1
119899 119899 ge 1 (45)
Combining the result of adding (43) and (44) with (45) weobtain
1199063119899minus1
= minus1
2(11990610158401015840
1119899minus1+ 11990610158401015840
2119899minus1) 1199092
+3
2(1199061119899minus1
+ 1199062119899minus1
) 119899 ge 1
(46)
The substitution of (46) into (43) and (44) respectively leadsus to
1199061119899=
1
2119899(1199092
11990610158401015840
1119899minus1minus 31199061119899minus1
+ 31199062119899minus1
minus1199092
11990610158401015840
2119899minus1+ 2 (minus1)
119899minus1
1199092
) 119899 ge 1
1199062119899=
1
2119899(1199092
11990610158401015840
2119899minus1minus 31199062119899minus1
+ 31199061119899minus1
minus1199092
11990610158401015840
1119899minus1+ 2 (minus1)
119899minus1
1199092
) 119899 ge 1
(47)
Discrete Dynamics in Nature and Society 5
From recursion formulas (46) and (47) we get the functions
11990610(119909) = 0 119906
11(119909) = 119909
2
11990612(119909) =
minus1199092
2
11990613=1199092
3 119906
14=minus1199092
4sdot sdot sdot
(48)
11990620(119909) = 0 119906
21(119909) = 119909
2
11990622(119909) =
minus1199092
2
11990623=1199092
3 119906
24=minus1199092
4sdot sdot sdot
(49)
11990630(119909) = 0 119906
31(119909) = 119909
2
11990632(119909) =
minus1199092
2
11990633=1199092
3 119906
34=minus1199092
4sdot sdot sdot
(50)
After substituting (48) through (50) into series (35) (36) and(37) respectively we get
1199061(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (51)
1199062(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (52)
1199063(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (53)
After identifying the 119899th terms of the series (51) (52) and (53)as ((minus1)119899minus1119899)119905119899 we conclude that
1199061(119905 119909) = 119909
2 ln (1 + 119905)
1199062(119905 119909) = 119909
2 ln (1 + 119905)
1199063(119905 119909) = 119909
2 ln (1 + 119905)
(54)
which is the exact solution of (29)ndash(32) (see (39))
43 Nonlinear Index-Two SPDAE with Variable Coefficients(1198981= 2 119898
2= 1) Finally consider the following
1199061119905= 119891 (119909) 119906
1119909119909+ 11990611199061119909minus1 minus 119905
1 + 1199051199063 (55)
1199062119905= 119892 (119909) 119906
2119909119909minus 11990621199062119909+1 + 119905
1 minus 1199051199063 (56)
0 = 1199061(1 + 119905) minus 119906
2(1 minus 119905) minusinfin lt 119909 lt infin minus1 lt 119905 lt 1
(57)
subject to the initial conditions
1199061(0 119909) = 119909 119906
2(0 119909) = 119909 119906
3(0 119909) = 2119909 (58)
where 119891(119909) and 119892(119909) are analytical functions on minusinfin lt 119909 lt
infin
The integration of (55) and (56)with respect to 119905 andusingthe initial conditions (58) lead to
1199061(119905 119909) = 119909 + int
119905
0
[119891 (119909) 1199061119909119909
+ 11990611199061119909minus1 minus 119905
1 + 1199051199063] 119889119905 (59)
1199062(119905 119909) = 119909 + int
119905
0
[119892 (119909) 1199062119909119909
minus 11990621199062119909+1 + 119905
1 minus 1199051199063] 119889119905 (60)
PSM assumes once again that 1199061(119905 119909) 119906
2(119905 119909) and 119906
3(119905 119909)
can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (61)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (62)
1199063(119905 119909) = 119906
30(119909) + 119906
31(119909) 119905 + 119906
32(119909) 1199052
+ sdot sdot sdot (63)
where 11990610(119909) 11990611(119909) 119906
20(119909) 11990621(119909) 119906
30(119909) 11990631(119909)
are unknown functions to be determined later on by thePSM method
Substituting (61) and (63) into (59) and also (62) and (63)into (60) respectively we getinfin
sum
119899=0
1199061119899119905119899
= 119909 + int
119905
0
119891 (119909)
infin
sum
119899=0
11990610158401015840
1119899119905119899
119889119905 + int
119905
0
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898119905119899+119898
119889119905
minus int
119905
0
(1 minus 119905)
infin
sum
119899=0
infin
sum
119898=0
(minus1)119899
1199063119898119905119899+119898
119889119905
(64)infin
sum
119899=0
1199062119899119905119899
= 119909 + int
119905
0
119892 (119909)
infin
sum
119899=0
11990610158401015840
2119899119905119899
119889119905 + int
119905
0
infin
sum
119899=0
infin
sum
119898=0
11990621198991199061015840
2119898119905119899+119898
119889119905
minus int
119905
0
(1 + 119905)
infin
sum
119899=0
infin
sum
119898=0
1199063119898119905119899+119898
119889119905
(65)
where we have employed the Taylor series expansions
1
1 minus 119905=
infin
sum
119899=0
119905119899
1
1 + 119905=
infin
sum
119899=0
(minus1)119899
119905119899
(66)
After integrating and standardizing the summation indexwe get the following recursion formulas from (64) and (65)respectively
minus 11990610+ 119909 minus 119906
30119905 minus
1
2(11990631minus 211990630) 1199052
minus1
3(11990632minus 211990631+ 211990630) 1199053
minus1
4(11990633minus 211990632+ 211990631minus 211990630) 1199054
+
infin
sum
119896=1
[119891 (119909) 119906
10158401015840
1119896minus1
119896+
infin
sum
119898=0
1199061015840
11198981199061119896minus119898minus1
119896minus 1199061119896] 119905119896
= 0
minus 11990620+ 119909 + 119906
30119905 +
1
2(11990631+ 211990630) 1199052
+1
3(11990632+ 211990631+ 211990630) 1199053
6 Discrete Dynamics in Nature and Society
+1
4(11990633+ 211990632+ 211990631+ 211990630) 1199054
+
infin
sum
119896=1
[119892 (119909) 119906
10158401015840
2119896minus1
119896minus
infin
sum
119898=0
1199061015840
21198981199062119896minus119898minus1
119896minus 1199062119896] 119905119896
= 0
(67)
From (57) we obtain
infin
sum
119898=0
1199062119898119905119898
= (1 + 119905)
infin
sum
119899=0
infin
sum
119895=0
119905119899+119895
1199061119895 (68)
after using again the first series of (66)After standardizing the summation index we get a third
recurrence formula from (68)
1199062119896=
infin
sum
119899=0
[1199061119896minus119899
+ 1199061119896minus119899minus1
] where 119896 = 0 1 2 3
(69)
From recursion formulas (67) and (69) we get the followingcoupled equations
11990610= 11990610(0 119909) (70)
11990611= 119891 (119909) 119906
10158401015840
10+ 1199061015840
1011990610minus 11990630 (71)
11990612= 119891 (119909)
11990610158401015840
11
2+1199061015840
1011990611+ 1199061015840
1111990610
2+11990631
2 (72)
11990613= 119891 (119909)
11990610158401015840
12
3+1199061015840
1011990612+ 1199061015840
1111990611+ 1199061015840
1211990610
3
minus11990632+ 211990630minus 211990631
3
(73)
11990614=11990610158401015840
13
4+1199061015840
1011990613+ 1199061015840
1111990612+ 1199061015840
1211990611+ 1199061015840
1311990610
4
minus11990633minus 211990632+ 211990631minus 211990630
4
(74)
11990620= 11990620(0 119909) (75)
11990621= 119892 (119909) 119906
10158401015840
20minus 1199061015840
2011990620+ 11990630 (76)
11990622= 119892 (119909)
11990610158401015840
21
2minus1199061015840
2011990621+ 1199061015840
2111990620
2+11990631+ 211990630
2 (77)
11990623= 119892 (119909)
11990610158401015840
22
3minus1199061015840
2011990622+ 1199061015840
2111990621+ 1199061015840
2211990620
3
+11990632+ 211990630+ 211990631
3
(78)
11990624= 119892 (119909)
11990610158401015840
23
4minus1199061015840
2011990623+ 1199061015840
2111990622+ 1199061015840
2211990621+ 1199061015840
2311990620
4
+11990633+ 211990632+ 211990631+ 211990630
4
(79)
11990620= 11990610 (80)
11990621= 211990610+ 11990611 (81)
11990622= 11990612+ 211990611+ 211990610 (82)
11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)
11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610
(84)
From (70) through (84) we get the functions
11990610= 119909 119906
11= minus119909 119906
12= 119909
11990613= minus119909 119906
14= 119909 sdot sdot sdot
(85)
11990620= 119909 119906
21= 119909 119906
22= 119909
11990623= 119909 119906
24= 119909 sdot sdot sdot
(86)
11990630= 2119909 119906
31= 0 119906
32= 2119909
11990633= 0 119906
34= 2119909
(87)
Substituting (85) through (87) into series (61) (62) and (63)respectively we get
1199061(119905 119909) = 119909 (1 minus 119905 + 119905
2
minus 1199053
+ 1199054
+ sdot sdot sdot ) (88)
1199062(119905 119909) = 119909 (1 + 119905 + 119905
2
+ 1199053
+ 1199054
+ sdot sdot sdot ) (89)
1199063(119905 119909) = 2119909 (1 + 119905
2
+ 1199054
+ 1199056
+ sdot sdot sdot ) (90)
After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms
1199061(119905 119909) =
119909
1 + 119905
1199062(119905 119909) =
119909
1 minus 119905
1199063(119905 119909) =
2119909
1 minus 1199052
(91)
which is the exact solution of (55)ndash(58) where we haveemployed (66) and
1
1 minus 1199052=
infin
sum
119899=0
1199052119899
(92)
This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general
Discrete Dynamics in Nature and Society 7
terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)
5 Discussion
In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution
Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs
One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs
Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems
6 Conclusion
By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above
all of index greater than one combining PSM and Laplace-Pade resummation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024
References
[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998
[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999
[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000
[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007
[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007
[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007
[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[11] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional
8 Discrete Dynamics in Nature and Society
decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011
[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011
[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006
[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008
[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009
[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005
[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009
[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008
[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011
[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012
[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012
[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013
[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014
[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012
[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012
[42] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011
[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011
[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra
Discrete Dynamics in Nature and Society 9
and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012
[46] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011
[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956
[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906
[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995
[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995
[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013
[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013
[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014
[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011
[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Discrete Dynamics in Nature and Society
the substitution of (20) (21) (22) (23) and (24) leads to
11990615= 0 (25)
in the same way we obtain
11990616= 11990617= 11990618= sdot sdot sdot = 0 (26)
Substituting (20) through (26) into (14) leads us to
1199061(119905 119909) = 119909
3
119905 (27)
Finally substituting (27) into (11) leads to
1199062(119905 119909) =
1199054
119909 (28)
Thus (27) and (28) are the exact solution for SPDAE system(10)ndash(12)
42 Linear Index-Two SPDAEwithVariable Coefficients (1198981=
2 1198982= 1) Consider the following
1199061119905= 1199092
1199061119909119909
minus 31199061+ 1199063+
1199092
1 + 119905 (29)
1199062119905= 1199092
1199062119909119909
minus 31199062+ 1199063+
1199092
1 + 119905 (30)
0 = 1199061+ 1199062minus 21199092 ln (1 + 119905) (31)
subject to the initial conditions
1199061(0 119909) = 0 119906
2(0 119909) = 0
minus1 lt 119905 le 1 minusinfin lt 119909 lt infin
(32)
The integration of (29) and (30) with respect to 119905 and usingthe initial conditions (32) lead to
1199061(119905 119909) = int
119905
0
[1199092
1199061119909119909
minus 31199061+ 1199063] 119889119905 + 119909
2 ln (1 + 119905) (33)
1199062(119905 119909) = int
119905
0
[1199092
1199062119909119909
minus 31199062+ 1199063] 119889119905 + 119909
2 ln (1 + 119905) (34)
assuming that 1199061(119905 119909) 119906
2(119905 119909) and 119906
3(119905 119909) can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (35)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (36)
1199063(119905 119909) = 119906
30(119909) + 119906
31(119909) 119905 + 119906
32(119909) 1199052
+ sdot sdot sdot (37)
where 11990610(119909) 11990611(119909) 119906
20(119909) 11990621(119909) 119906
30(119909) 11990631(119909)
are unknown functions to be determined later on by thePSM method
After substituting (35) and (37) into (33) we get
119906101199050
+
infin
sum
119896=1
1
119896[1198961199061119896minus 1199092
11990610158401015840
1119896minus1+ 31199061119896minus1
minus1199063119896minus1
minus 1199092
(minus1)119896minus1
] 119905119896
= 0
(38)
where we have standardized the summation index and em-ployed the following Taylor series expansion
ln (1 + 119905) =infin
sum
119899=1
(minus1)119899minus1
119899119905119899
minus1 lt 119905 le 1 (39)
In the same way the substitution of (36) and (37) into (34)leads to
119906201199050
+
infin
sum
119896=1
1
119896[1198961199062119896minus 1199092
11990610158401015840
2119896minus1+ 31199062119896minus1
minus1199063119896minus1
minus 1199092
(minus1)119896minus1
] 119905119896
= 0
(40)
On the other hand after substituting (35) (36) and (39) into(31) we have
infin
sum
119896=1
[1199061119896+ 1199062119896minus21199092
119896(minus1)119896minus1
] 119905119896
= 0 (41)
where we have employed the following results deduced from(38) and (40)
11990610= 11990620= 0 (42)
Equations (38) (40) and (41) give rise to the followingformulas
1199061119899=1199092
11990610158401015840
1119899minus1minus 31199061119899minus1
+ 1199063119899minus1
+ (minus1)119899minus1
1199092
119899 119899 ge 1 (43)
1199062119899=1199092
11990610158401015840
2119899minus1minus 31199062119899minus1
+ 1199063119899minus1
+ (minus1)119899minus1
1199092
119899 119899 ge 1 (44)
1199061119899+ 1199062119899=21199092
(minus1)119899minus1
119899 119899 ge 1 (45)
Combining the result of adding (43) and (44) with (45) weobtain
1199063119899minus1
= minus1
2(11990610158401015840
1119899minus1+ 11990610158401015840
2119899minus1) 1199092
+3
2(1199061119899minus1
+ 1199062119899minus1
) 119899 ge 1
(46)
The substitution of (46) into (43) and (44) respectively leadsus to
1199061119899=
1
2119899(1199092
11990610158401015840
1119899minus1minus 31199061119899minus1
+ 31199062119899minus1
minus1199092
11990610158401015840
2119899minus1+ 2 (minus1)
119899minus1
1199092
) 119899 ge 1
1199062119899=
1
2119899(1199092
11990610158401015840
2119899minus1minus 31199062119899minus1
+ 31199061119899minus1
minus1199092
11990610158401015840
1119899minus1+ 2 (minus1)
119899minus1
1199092
) 119899 ge 1
(47)
Discrete Dynamics in Nature and Society 5
From recursion formulas (46) and (47) we get the functions
11990610(119909) = 0 119906
11(119909) = 119909
2
11990612(119909) =
minus1199092
2
11990613=1199092
3 119906
14=minus1199092
4sdot sdot sdot
(48)
11990620(119909) = 0 119906
21(119909) = 119909
2
11990622(119909) =
minus1199092
2
11990623=1199092
3 119906
24=minus1199092
4sdot sdot sdot
(49)
11990630(119909) = 0 119906
31(119909) = 119909
2
11990632(119909) =
minus1199092
2
11990633=1199092
3 119906
34=minus1199092
4sdot sdot sdot
(50)
After substituting (48) through (50) into series (35) (36) and(37) respectively we get
1199061(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (51)
1199062(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (52)
1199063(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (53)
After identifying the 119899th terms of the series (51) (52) and (53)as ((minus1)119899minus1119899)119905119899 we conclude that
1199061(119905 119909) = 119909
2 ln (1 + 119905)
1199062(119905 119909) = 119909
2 ln (1 + 119905)
1199063(119905 119909) = 119909
2 ln (1 + 119905)
(54)
which is the exact solution of (29)ndash(32) (see (39))
43 Nonlinear Index-Two SPDAE with Variable Coefficients(1198981= 2 119898
2= 1) Finally consider the following
1199061119905= 119891 (119909) 119906
1119909119909+ 11990611199061119909minus1 minus 119905
1 + 1199051199063 (55)
1199062119905= 119892 (119909) 119906
2119909119909minus 11990621199062119909+1 + 119905
1 minus 1199051199063 (56)
0 = 1199061(1 + 119905) minus 119906
2(1 minus 119905) minusinfin lt 119909 lt infin minus1 lt 119905 lt 1
(57)
subject to the initial conditions
1199061(0 119909) = 119909 119906
2(0 119909) = 119909 119906
3(0 119909) = 2119909 (58)
where 119891(119909) and 119892(119909) are analytical functions on minusinfin lt 119909 lt
infin
The integration of (55) and (56)with respect to 119905 andusingthe initial conditions (58) lead to
1199061(119905 119909) = 119909 + int
119905
0
[119891 (119909) 1199061119909119909
+ 11990611199061119909minus1 minus 119905
1 + 1199051199063] 119889119905 (59)
1199062(119905 119909) = 119909 + int
119905
0
[119892 (119909) 1199062119909119909
minus 11990621199062119909+1 + 119905
1 minus 1199051199063] 119889119905 (60)
PSM assumes once again that 1199061(119905 119909) 119906
2(119905 119909) and 119906
3(119905 119909)
can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (61)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (62)
1199063(119905 119909) = 119906
30(119909) + 119906
31(119909) 119905 + 119906
32(119909) 1199052
+ sdot sdot sdot (63)
where 11990610(119909) 11990611(119909) 119906
20(119909) 11990621(119909) 119906
30(119909) 11990631(119909)
are unknown functions to be determined later on by thePSM method
Substituting (61) and (63) into (59) and also (62) and (63)into (60) respectively we getinfin
sum
119899=0
1199061119899119905119899
= 119909 + int
119905
0
119891 (119909)
infin
sum
119899=0
11990610158401015840
1119899119905119899
119889119905 + int
119905
0
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898119905119899+119898
119889119905
minus int
119905
0
(1 minus 119905)
infin
sum
119899=0
infin
sum
119898=0
(minus1)119899
1199063119898119905119899+119898
119889119905
(64)infin
sum
119899=0
1199062119899119905119899
= 119909 + int
119905
0
119892 (119909)
infin
sum
119899=0
11990610158401015840
2119899119905119899
119889119905 + int
119905
0
infin
sum
119899=0
infin
sum
119898=0
11990621198991199061015840
2119898119905119899+119898
119889119905
minus int
119905
0
(1 + 119905)
infin
sum
119899=0
infin
sum
119898=0
1199063119898119905119899+119898
119889119905
(65)
where we have employed the Taylor series expansions
1
1 minus 119905=
infin
sum
119899=0
119905119899
1
1 + 119905=
infin
sum
119899=0
(minus1)119899
119905119899
(66)
After integrating and standardizing the summation indexwe get the following recursion formulas from (64) and (65)respectively
minus 11990610+ 119909 minus 119906
30119905 minus
1
2(11990631minus 211990630) 1199052
minus1
3(11990632minus 211990631+ 211990630) 1199053
minus1
4(11990633minus 211990632+ 211990631minus 211990630) 1199054
+
infin
sum
119896=1
[119891 (119909) 119906
10158401015840
1119896minus1
119896+
infin
sum
119898=0
1199061015840
11198981199061119896minus119898minus1
119896minus 1199061119896] 119905119896
= 0
minus 11990620+ 119909 + 119906
30119905 +
1
2(11990631+ 211990630) 1199052
+1
3(11990632+ 211990631+ 211990630) 1199053
6 Discrete Dynamics in Nature and Society
+1
4(11990633+ 211990632+ 211990631+ 211990630) 1199054
+
infin
sum
119896=1
[119892 (119909) 119906
10158401015840
2119896minus1
119896minus
infin
sum
119898=0
1199061015840
21198981199062119896minus119898minus1
119896minus 1199062119896] 119905119896
= 0
(67)
From (57) we obtain
infin
sum
119898=0
1199062119898119905119898
= (1 + 119905)
infin
sum
119899=0
infin
sum
119895=0
119905119899+119895
1199061119895 (68)
after using again the first series of (66)After standardizing the summation index we get a third
recurrence formula from (68)
1199062119896=
infin
sum
119899=0
[1199061119896minus119899
+ 1199061119896minus119899minus1
] where 119896 = 0 1 2 3
(69)
From recursion formulas (67) and (69) we get the followingcoupled equations
11990610= 11990610(0 119909) (70)
11990611= 119891 (119909) 119906
10158401015840
10+ 1199061015840
1011990610minus 11990630 (71)
11990612= 119891 (119909)
11990610158401015840
11
2+1199061015840
1011990611+ 1199061015840
1111990610
2+11990631
2 (72)
11990613= 119891 (119909)
11990610158401015840
12
3+1199061015840
1011990612+ 1199061015840
1111990611+ 1199061015840
1211990610
3
minus11990632+ 211990630minus 211990631
3
(73)
11990614=11990610158401015840
13
4+1199061015840
1011990613+ 1199061015840
1111990612+ 1199061015840
1211990611+ 1199061015840
1311990610
4
minus11990633minus 211990632+ 211990631minus 211990630
4
(74)
11990620= 11990620(0 119909) (75)
11990621= 119892 (119909) 119906
10158401015840
20minus 1199061015840
2011990620+ 11990630 (76)
11990622= 119892 (119909)
11990610158401015840
21
2minus1199061015840
2011990621+ 1199061015840
2111990620
2+11990631+ 211990630
2 (77)
11990623= 119892 (119909)
11990610158401015840
22
3minus1199061015840
2011990622+ 1199061015840
2111990621+ 1199061015840
2211990620
3
+11990632+ 211990630+ 211990631
3
(78)
11990624= 119892 (119909)
11990610158401015840
23
4minus1199061015840
2011990623+ 1199061015840
2111990622+ 1199061015840
2211990621+ 1199061015840
2311990620
4
+11990633+ 211990632+ 211990631+ 211990630
4
(79)
11990620= 11990610 (80)
11990621= 211990610+ 11990611 (81)
11990622= 11990612+ 211990611+ 211990610 (82)
11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)
11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610
(84)
From (70) through (84) we get the functions
11990610= 119909 119906
11= minus119909 119906
12= 119909
11990613= minus119909 119906
14= 119909 sdot sdot sdot
(85)
11990620= 119909 119906
21= 119909 119906
22= 119909
11990623= 119909 119906
24= 119909 sdot sdot sdot
(86)
11990630= 2119909 119906
31= 0 119906
32= 2119909
11990633= 0 119906
34= 2119909
(87)
Substituting (85) through (87) into series (61) (62) and (63)respectively we get
1199061(119905 119909) = 119909 (1 minus 119905 + 119905
2
minus 1199053
+ 1199054
+ sdot sdot sdot ) (88)
1199062(119905 119909) = 119909 (1 + 119905 + 119905
2
+ 1199053
+ 1199054
+ sdot sdot sdot ) (89)
1199063(119905 119909) = 2119909 (1 + 119905
2
+ 1199054
+ 1199056
+ sdot sdot sdot ) (90)
After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms
1199061(119905 119909) =
119909
1 + 119905
1199062(119905 119909) =
119909
1 minus 119905
1199063(119905 119909) =
2119909
1 minus 1199052
(91)
which is the exact solution of (55)ndash(58) where we haveemployed (66) and
1
1 minus 1199052=
infin
sum
119899=0
1199052119899
(92)
This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general
Discrete Dynamics in Nature and Society 7
terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)
5 Discussion
In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution
Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs
One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs
Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems
6 Conclusion
By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above
all of index greater than one combining PSM and Laplace-Pade resummation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024
References
[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998
[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999
[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000
[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007
[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007
[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007
[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[11] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional
8 Discrete Dynamics in Nature and Society
decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011
[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011
[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006
[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008
[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009
[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005
[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009
[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008
[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011
[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012
[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012
[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013
[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014
[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012
[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012
[42] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011
[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011
[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra
Discrete Dynamics in Nature and Society 9
and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012
[46] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011
[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956
[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906
[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995
[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995
[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013
[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013
[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014
[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011
[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 5
From recursion formulas (46) and (47) we get the functions
11990610(119909) = 0 119906
11(119909) = 119909
2
11990612(119909) =
minus1199092
2
11990613=1199092
3 119906
14=minus1199092
4sdot sdot sdot
(48)
11990620(119909) = 0 119906
21(119909) = 119909
2
11990622(119909) =
minus1199092
2
11990623=1199092
3 119906
24=minus1199092
4sdot sdot sdot
(49)
11990630(119909) = 0 119906
31(119909) = 119909
2
11990632(119909) =
minus1199092
2
11990633=1199092
3 119906
34=minus1199092
4sdot sdot sdot
(50)
After substituting (48) through (50) into series (35) (36) and(37) respectively we get
1199061(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (51)
1199062(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (52)
1199063(119905 119909) = 119909
2
[119905 minus1199052
2+1199053
3minus1199054
4+ sdot sdot sdot ] (53)
After identifying the 119899th terms of the series (51) (52) and (53)as ((minus1)119899minus1119899)119905119899 we conclude that
1199061(119905 119909) = 119909
2 ln (1 + 119905)
1199062(119905 119909) = 119909
2 ln (1 + 119905)
1199063(119905 119909) = 119909
2 ln (1 + 119905)
(54)
which is the exact solution of (29)ndash(32) (see (39))
43 Nonlinear Index-Two SPDAE with Variable Coefficients(1198981= 2 119898
2= 1) Finally consider the following
1199061119905= 119891 (119909) 119906
1119909119909+ 11990611199061119909minus1 minus 119905
1 + 1199051199063 (55)
1199062119905= 119892 (119909) 119906
2119909119909minus 11990621199062119909+1 + 119905
1 minus 1199051199063 (56)
0 = 1199061(1 + 119905) minus 119906
2(1 minus 119905) minusinfin lt 119909 lt infin minus1 lt 119905 lt 1
(57)
subject to the initial conditions
1199061(0 119909) = 119909 119906
2(0 119909) = 119909 119906
3(0 119909) = 2119909 (58)
where 119891(119909) and 119892(119909) are analytical functions on minusinfin lt 119909 lt
infin
The integration of (55) and (56)with respect to 119905 andusingthe initial conditions (58) lead to
1199061(119905 119909) = 119909 + int
119905
0
[119891 (119909) 1199061119909119909
+ 11990611199061119909minus1 minus 119905
1 + 1199051199063] 119889119905 (59)
1199062(119905 119909) = 119909 + int
119905
0
[119892 (119909) 1199062119909119909
minus 11990621199062119909+1 + 119905
1 minus 1199051199063] 119889119905 (60)
PSM assumes once again that 1199061(119905 119909) 119906
2(119905 119909) and 119906
3(119905 119909)
can be written as
1199061(119905 119909) = 119906
10(119909) + 119906
11(119909) 119905 + 119906
12(119909) 1199052
+ sdot sdot sdot (61)
1199062(119905 119909) = 119906
20(119909) + 119906
21(119909) 119905 + 119906
22(119909) 1199052
+ sdot sdot sdot (62)
1199063(119905 119909) = 119906
30(119909) + 119906
31(119909) 119905 + 119906
32(119909) 1199052
+ sdot sdot sdot (63)
where 11990610(119909) 11990611(119909) 119906
20(119909) 11990621(119909) 119906
30(119909) 11990631(119909)
are unknown functions to be determined later on by thePSM method
Substituting (61) and (63) into (59) and also (62) and (63)into (60) respectively we getinfin
sum
119899=0
1199061119899119905119899
= 119909 + int
119905
0
119891 (119909)
infin
sum
119899=0
11990610158401015840
1119899119905119899
119889119905 + int
119905
0
infin
sum
119899=0
infin
sum
119898=0
11990611198991199061015840
1119898119905119899+119898
119889119905
minus int
119905
0
(1 minus 119905)
infin
sum
119899=0
infin
sum
119898=0
(minus1)119899
1199063119898119905119899+119898
119889119905
(64)infin
sum
119899=0
1199062119899119905119899
= 119909 + int
119905
0
119892 (119909)
infin
sum
119899=0
11990610158401015840
2119899119905119899
119889119905 + int
119905
0
infin
sum
119899=0
infin
sum
119898=0
11990621198991199061015840
2119898119905119899+119898
119889119905
minus int
119905
0
(1 + 119905)
infin
sum
119899=0
infin
sum
119898=0
1199063119898119905119899+119898
119889119905
(65)
where we have employed the Taylor series expansions
1
1 minus 119905=
infin
sum
119899=0
119905119899
1
1 + 119905=
infin
sum
119899=0
(minus1)119899
119905119899
(66)
After integrating and standardizing the summation indexwe get the following recursion formulas from (64) and (65)respectively
minus 11990610+ 119909 minus 119906
30119905 minus
1
2(11990631minus 211990630) 1199052
minus1
3(11990632minus 211990631+ 211990630) 1199053
minus1
4(11990633minus 211990632+ 211990631minus 211990630) 1199054
+
infin
sum
119896=1
[119891 (119909) 119906
10158401015840
1119896minus1
119896+
infin
sum
119898=0
1199061015840
11198981199061119896minus119898minus1
119896minus 1199061119896] 119905119896
= 0
minus 11990620+ 119909 + 119906
30119905 +
1
2(11990631+ 211990630) 1199052
+1
3(11990632+ 211990631+ 211990630) 1199053
6 Discrete Dynamics in Nature and Society
+1
4(11990633+ 211990632+ 211990631+ 211990630) 1199054
+
infin
sum
119896=1
[119892 (119909) 119906
10158401015840
2119896minus1
119896minus
infin
sum
119898=0
1199061015840
21198981199062119896minus119898minus1
119896minus 1199062119896] 119905119896
= 0
(67)
From (57) we obtain
infin
sum
119898=0
1199062119898119905119898
= (1 + 119905)
infin
sum
119899=0
infin
sum
119895=0
119905119899+119895
1199061119895 (68)
after using again the first series of (66)After standardizing the summation index we get a third
recurrence formula from (68)
1199062119896=
infin
sum
119899=0
[1199061119896minus119899
+ 1199061119896minus119899minus1
] where 119896 = 0 1 2 3
(69)
From recursion formulas (67) and (69) we get the followingcoupled equations
11990610= 11990610(0 119909) (70)
11990611= 119891 (119909) 119906
10158401015840
10+ 1199061015840
1011990610minus 11990630 (71)
11990612= 119891 (119909)
11990610158401015840
11
2+1199061015840
1011990611+ 1199061015840
1111990610
2+11990631
2 (72)
11990613= 119891 (119909)
11990610158401015840
12
3+1199061015840
1011990612+ 1199061015840
1111990611+ 1199061015840
1211990610
3
minus11990632+ 211990630minus 211990631
3
(73)
11990614=11990610158401015840
13
4+1199061015840
1011990613+ 1199061015840
1111990612+ 1199061015840
1211990611+ 1199061015840
1311990610
4
minus11990633minus 211990632+ 211990631minus 211990630
4
(74)
11990620= 11990620(0 119909) (75)
11990621= 119892 (119909) 119906
10158401015840
20minus 1199061015840
2011990620+ 11990630 (76)
11990622= 119892 (119909)
11990610158401015840
21
2minus1199061015840
2011990621+ 1199061015840
2111990620
2+11990631+ 211990630
2 (77)
11990623= 119892 (119909)
11990610158401015840
22
3minus1199061015840
2011990622+ 1199061015840
2111990621+ 1199061015840
2211990620
3
+11990632+ 211990630+ 211990631
3
(78)
11990624= 119892 (119909)
11990610158401015840
23
4minus1199061015840
2011990623+ 1199061015840
2111990622+ 1199061015840
2211990621+ 1199061015840
2311990620
4
+11990633+ 211990632+ 211990631+ 211990630
4
(79)
11990620= 11990610 (80)
11990621= 211990610+ 11990611 (81)
11990622= 11990612+ 211990611+ 211990610 (82)
11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)
11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610
(84)
From (70) through (84) we get the functions
11990610= 119909 119906
11= minus119909 119906
12= 119909
11990613= minus119909 119906
14= 119909 sdot sdot sdot
(85)
11990620= 119909 119906
21= 119909 119906
22= 119909
11990623= 119909 119906
24= 119909 sdot sdot sdot
(86)
11990630= 2119909 119906
31= 0 119906
32= 2119909
11990633= 0 119906
34= 2119909
(87)
Substituting (85) through (87) into series (61) (62) and (63)respectively we get
1199061(119905 119909) = 119909 (1 minus 119905 + 119905
2
minus 1199053
+ 1199054
+ sdot sdot sdot ) (88)
1199062(119905 119909) = 119909 (1 + 119905 + 119905
2
+ 1199053
+ 1199054
+ sdot sdot sdot ) (89)
1199063(119905 119909) = 2119909 (1 + 119905
2
+ 1199054
+ 1199056
+ sdot sdot sdot ) (90)
After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms
1199061(119905 119909) =
119909
1 + 119905
1199062(119905 119909) =
119909
1 minus 119905
1199063(119905 119909) =
2119909
1 minus 1199052
(91)
which is the exact solution of (55)ndash(58) where we haveemployed (66) and
1
1 minus 1199052=
infin
sum
119899=0
1199052119899
(92)
This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general
Discrete Dynamics in Nature and Society 7
terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)
5 Discussion
In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution
Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs
One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs
Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems
6 Conclusion
By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above
all of index greater than one combining PSM and Laplace-Pade resummation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024
References
[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998
[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999
[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000
[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007
[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007
[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007
[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[11] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional
8 Discrete Dynamics in Nature and Society
decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011
[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011
[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006
[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008
[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009
[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005
[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009
[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008
[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011
[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012
[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012
[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013
[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014
[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012
[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012
[42] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011
[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011
[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra
Discrete Dynamics in Nature and Society 9
and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012
[46] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011
[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956
[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906
[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995
[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995
[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013
[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013
[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014
[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011
[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Discrete Dynamics in Nature and Society
+1
4(11990633+ 211990632+ 211990631+ 211990630) 1199054
+
infin
sum
119896=1
[119892 (119909) 119906
10158401015840
2119896minus1
119896minus
infin
sum
119898=0
1199061015840
21198981199062119896minus119898minus1
119896minus 1199062119896] 119905119896
= 0
(67)
From (57) we obtain
infin
sum
119898=0
1199062119898119905119898
= (1 + 119905)
infin
sum
119899=0
infin
sum
119895=0
119905119899+119895
1199061119895 (68)
after using again the first series of (66)After standardizing the summation index we get a third
recurrence formula from (68)
1199062119896=
infin
sum
119899=0
[1199061119896minus119899
+ 1199061119896minus119899minus1
] where 119896 = 0 1 2 3
(69)
From recursion formulas (67) and (69) we get the followingcoupled equations
11990610= 11990610(0 119909) (70)
11990611= 119891 (119909) 119906
10158401015840
10+ 1199061015840
1011990610minus 11990630 (71)
11990612= 119891 (119909)
11990610158401015840
11
2+1199061015840
1011990611+ 1199061015840
1111990610
2+11990631
2 (72)
11990613= 119891 (119909)
11990610158401015840
12
3+1199061015840
1011990612+ 1199061015840
1111990611+ 1199061015840
1211990610
3
minus11990632+ 211990630minus 211990631
3
(73)
11990614=11990610158401015840
13
4+1199061015840
1011990613+ 1199061015840
1111990612+ 1199061015840
1211990611+ 1199061015840
1311990610
4
minus11990633minus 211990632+ 211990631minus 211990630
4
(74)
11990620= 11990620(0 119909) (75)
11990621= 119892 (119909) 119906
10158401015840
20minus 1199061015840
2011990620+ 11990630 (76)
11990622= 119892 (119909)
11990610158401015840
21
2minus1199061015840
2011990621+ 1199061015840
2111990620
2+11990631+ 211990630
2 (77)
11990623= 119892 (119909)
11990610158401015840
22
3minus1199061015840
2011990622+ 1199061015840
2111990621+ 1199061015840
2211990620
3
+11990632+ 211990630+ 211990631
3
(78)
11990624= 119892 (119909)
11990610158401015840
23
4minus1199061015840
2011990623+ 1199061015840
2111990622+ 1199061015840
2211990621+ 1199061015840
2311990620
4
+11990633+ 211990632+ 211990631+ 211990630
4
(79)
11990620= 11990610 (80)
11990621= 211990610+ 11990611 (81)
11990622= 11990612+ 211990611+ 211990610 (82)
11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)
11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610
(84)
From (70) through (84) we get the functions
11990610= 119909 119906
11= minus119909 119906
12= 119909
11990613= minus119909 119906
14= 119909 sdot sdot sdot
(85)
11990620= 119909 119906
21= 119909 119906
22= 119909
11990623= 119909 119906
24= 119909 sdot sdot sdot
(86)
11990630= 2119909 119906
31= 0 119906
32= 2119909
11990633= 0 119906
34= 2119909
(87)
Substituting (85) through (87) into series (61) (62) and (63)respectively we get
1199061(119905 119909) = 119909 (1 minus 119905 + 119905
2
minus 1199053
+ 1199054
+ sdot sdot sdot ) (88)
1199062(119905 119909) = 119909 (1 + 119905 + 119905
2
+ 1199053
+ 1199054
+ sdot sdot sdot ) (89)
1199063(119905 119909) = 2119909 (1 + 119905
2
+ 1199054
+ 1199056
+ sdot sdot sdot ) (90)
After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms
1199061(119905 119909) =
119909
1 + 119905
1199062(119905 119909) =
119909
1 minus 119905
1199063(119905 119909) =
2119909
1 minus 1199052
(91)
which is the exact solution of (55)ndash(58) where we haveemployed (66) and
1
1 minus 1199052=
infin
sum
119899=0
1199052119899
(92)
This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general
Discrete Dynamics in Nature and Society 7
terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)
5 Discussion
In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution
Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs
One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs
Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems
6 Conclusion
By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above
all of index greater than one combining PSM and Laplace-Pade resummation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024
References
[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998
[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999
[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000
[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007
[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007
[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007
[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[11] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional
8 Discrete Dynamics in Nature and Society
decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011
[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011
[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006
[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008
[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009
[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005
[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009
[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008
[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011
[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012
[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012
[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013
[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014
[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012
[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012
[42] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011
[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011
[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra
Discrete Dynamics in Nature and Society 9
and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012
[46] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011
[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956
[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906
[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995
[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995
[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013
[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013
[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014
[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011
[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 7
terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)
5 Discussion
In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution
Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs
One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs
Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems
6 Conclusion
By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above
all of index greater than one combining PSM and Laplace-Pade resummation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024
References
[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998
[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999
[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000
[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007
[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007
[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007
[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[11] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional
8 Discrete Dynamics in Nature and Society
decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011
[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011
[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006
[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008
[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009
[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005
[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009
[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008
[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011
[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012
[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012
[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013
[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014
[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012
[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012
[42] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011
[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011
[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra
Discrete Dynamics in Nature and Society 9
and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012
[46] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011
[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956
[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906
[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995
[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995
[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013
[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013
[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014
[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011
[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Discrete Dynamics in Nature and Society
decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011
[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011
[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006
[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008
[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009
[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005
[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009
[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008
[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011
[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012
[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012
[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013
[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014
[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012
[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012
[42] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011
[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011
[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra
Discrete Dynamics in Nature and Society 9
and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012
[46] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011
[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956
[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906
[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995
[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995
[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013
[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013
[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014
[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011
[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 9
and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012
[46] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011
[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956
[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906
[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995
[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995
[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013
[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013
[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014
[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011
[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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