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International Mathematical Forum, Vol. 7,2012, no. 13,625 - 630

A Solution for Nonlinear Systems of DifferentialEquations Using a Mixture of Elzaki Transform

And Differential Transform MethodTarig M. Elzaki

Mathematics Department, Faculty of Sciences and Arts-AlkamilKing Abdulaziz University, leddah Saudi Arabia

Math. Dept. Sudan University of Science and Technology-SudanTarig.alzaki@gmai1.com

AbstractIn this work, we define a new transforms called Elzaki transform, and differentialtransform method, and applied together to some nonlinear system of differentialequations to obtain their exact solutions. This method is based on Elzaki transform andthe nonlinear terms can be handled by the use of differential transform method.Keywords: Elzaki transform- differential transform method- nonlinear differentialequations- system of differential equations.

IntroductionIn the literature, there are numerous integral transformations methods widely used inphysics and astronomy as well as in engineering. The integral transform method is anefficient method for solving the linear systems of ordinary differential equations.Elzaki transform [1,2,3] is a useful technique for solving linear differential equations,however, Elzaki transform is totally incapable of handling nonlinear equations becauseof the difficulties that are caused by nonlinear terms. In this paper we use differentialtransform method [4,5,6] to decompose the nonlinear term, so that the solution can beobtained by iteration procedure.Elzaki transform:Consider functions in the set A defined by:

mailto:Tarig.alzaki@gmai1.commailto:Tarig.alzaki@gmai1.com

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626 T.M. Elzaki

Where M a constant is must be finite number and k . , k 2 can be finite or infinite.ELzaki transform denoted by the operator E O ' is defined by the integral equation:

00 tE [ r ( t ) J = T ( u ) = u f f ( t ) e - ; ; - d t , k j - : : ; ' u - : : ; ' k 2 , t ~ 0o

(1)

Theorem (1): [1]Let T ( u ) be ELzaki transform of f ( t ) [ E (r ( t ) ) = T ( u ) ] then:

(i) E [ f ' ( t ) ] = T ( u ) - u f ( O ) ( i i ) E [ J " ( t ) ] = T (~ ) - f ( O ) -u f ' ( O )u u

Proof:-(i) By the definition we have:

00 tE [ f ' ( t ) ] = u f f ' ( t ) e - ; d t , Integrating by parts, we get:o

E [ f ' ( t ) ] = T ( u ) - u f ( 0 )u(ii) Let g ( t ) = f ' ( t ) , Then: E [ g ' ( t ) ] = 2 _ E [ g ( t ) ] - u g ( 0 ) , using (i) to find that:u

E [ J " ( t ) ] = T ( ~ ) - f ( O ) - u f ' ( O )uDifferential Transform:Differential transform of the function y ( x ) is defined as follows:

y (k)~ ~ ! [ d ' ~ ~ x ) L (2)And the inverse differential transform of Y (k ) is defined as:

00y ( x ) = L Y ( k ) x kk~O

The main theorems of the one - dimensional differential transform are.Theorem (2):Theorem (3):

If w ( x ) = y ( x ) z ( x ) , then W ( k ) = Y ( k ) z ( x )If w ( x ) = cy ( x ) , Then W (k ) = cY (k )If w ( x ) = d y ( x ) , then W ( k ) = ( k + l ) Y ( k + 1 )d x

d n y ( x ) ( k + n ) !If w ( x ) = d x n ' then W ( k ) = k ! Y ( k + n )kIf w ( x ) =y ( x ) z ( x ) , then W ( x ) =L Y ( r ) Z ( k - r )

Theorem (4):

Theorem (5):

Theorem (6):

Theorem (7): If w (x ) =.x ", then W ( k ) ~ o ( k -n ) ~ f o k = nk - : f : - n

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Nonlinear systems of differential equations

Note that c is a constant and n is a nonnegative integerApplication:Example 1:Consider the following nonlinear system of differential equation:{Y;~= lOO~YI + IOOOY2Y2 -YI Y2(1+Y2)With the initial conditions:Y 1(0) = 1 , Y 2 (0) = 1Take Elzaki transform of (3) to get:

l YI~U )-uY I(O )=E [-I002YI + IOOOynY 2(U ) [ 2 J- --uYI(O )=E YI-Y2 -Y2UTake the inverse Elzaki transform of last equations to find:{y Ix ) =_1+ E - _I I{ UE[ - I 002y I+ I

2000Y n} Or we can write:

Y2(x )-I+E {UE[YI-Y2-Y2J}

{y In + 1 ) = E -I{uE [-1002 Y In ) + 1OOOAn]}Y 2 (n + 1) = E -I {U E [y I (n ) - Y 2 (n ) - An ]}

Where is a polynomial, and take it by differential transform method by:n

An = LY2(r)Y2(n -r)Since a series form of solution is

From the recursive relations (5) we have:

Y IIx ) = E -I{u E [-10 02+ lOOO]} = E -I(2u 3) = -2xY 21(x ) = E -I {u E [-I]} = E -I( _u 3) =-xY 12(x ) = E -I {uE [-1 0 0 2( - 2x ) + 1O OO ( 2x ) ]} = 2x 2

2Y 22 (x ) = E -I {uE [x ]} = ~2

The solution in a series form is given from equation (7) by:2Y I (x ) = 1- 2x + 2x 2 - .... = e -2x , Y 2 (x ) = 1- x + ~ - .... = e-x2

Example 2:Consider the Lotka-Volterra system which is an interacting species Predator-Preymodel governed by:

627

(3)

(4)

(5)

(6)

(7)

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628

j dY = y(l-x)d tdx-=x(l-y)d t

With the initial conditions:y (0) = 1 , x (0) = 2

Take Elzaki transform of eq (8), and making use of conditions (9) to find:

{y (u) =u2 +uE [y -xy]X (u) = 2u2 +uE [x -xY]

Then we can write recursive relations in the form:

{y (n + 1) = E -I{uE [y ( n ) - An]}X (n + 1) = E -I{uE [x ( n ) - An]}

nWhere An = Ly(r)x(n-r)From equations (10) and (11) we have:

Ao=2, y(1)=E-I{uE[I-2]}=-t ,x(I)=OAl =-2t , t2Y (2) =2' x (2) =t2

t3 t3y(3)=3 ' x(3)=-6Then the solution in a series form is given by:

t? t3 2 t3y(t)=I-t+-+-+ .... ,x(t)=2+t --+....233

T.M. Elzaki

(8)

(9)

(10)

(11)

Example 3:Consider the nonlinear system of differential equation (nonlinear reaction was takenfrom Hull).

dx-=-xd t

dy 2-=x-yd tdz 2-=yd t

With the initial conditions:x (0) = 0 , y (0) = 0, z (0) = 0

Take Elzaki transform of eq (12) and use eq (13) to find:

jx (u) = U 2 -uE (x)y (u) = uE [x - Y 2 ] 'Z (u) = uE (y 2)

(12)

(13)

Take the inverse Elzaki transform to find:

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Nonlinear systems of differential equations 629

x (t) = 1- E -I{uE (x ) }y (t) = E - I {uE [x - Y 2 J } ,Z (t) = E -I {uE (y 2) }

Then the recursive relations are given by:

x (n + 1) = - E -I{uE [x (n )]}y (n + 1) = E -I{uE [x ( n ) - A n ] }Z (n + 1) = E -I{uE (A n ) }

(14)

nWhere A n = Ly(r)y(n-r) (IS)From equations (14) and (IS) we have:x(1)=-E-1[uE(1)]=-E-1(U3)=-t , y(I)=E-1[uE(1)]=t , z(1)=O

t 2 t 2x (2) = -E -I[UE(-t)] =2 ' y (2) = E -I[uE (-t)] = -2' z (2) = 0For n = 2,3,4 we have:

t 3 t 3 t 3 t 4 St 4 t 4x(3)=-6' y(3)=-6 ' z(3)=3 ' x(4)= 24' y(4)=24' z(4)=-4

t5 t5 t5x(S)=-120 ' y(S)= 40 ' z(S)=- 60Then the solution in a series form is given by:i? t3 t4 t5x(t)=I-t +---+---+ ....2 6 24 120

t 2 t 3 St 4 t 5y(t)=t ----+-+-+....2 6 24 40t3 t4 t5z (t ) =-----+ .....3 4 60

ConclusionThe series solution for nonlinear systems of differential equations is obtained by usingElzaki transform and differential transform method. This technique is useful forsolving all nonlinear differential equations.Appendix:Elzaki transform of some Functions

f(t) E[t(t )]= r(u)1 2Ut 3Utn , n+2n. Uat 2e U--I-au

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630 T.M. Elzaki

sinai au.'l+a2u 2cosat 2u1+ a2u2

References[1] S. Islam, Yasir Khan, Naeem Faraz and Francis Austin, Numerical Solution ofLogistic Differential Equations by using the Laplace Decomposition Method, WorldApplied Sciences Journal 8 (9) :1100-1105,2010.[2] Nuran Guzel and Muhammet Nurulay, Solution of Shiff Systems By usingDifferential Transform Method, Dunlupinar universities Fen Bilimleri EnstitusuDergisi, 2008 ISSN 1302-3055, PP. 49-59.[3] Shin- Hsiang Chang, I-Ling Chang, A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Applied Mathematics andComputation 195 (2008) 799-808.[4] Tarig M. Elzaki, The New Integral Transform "Elzaki Transform" Global Journalof Pure and Applied Mathematics, ISSN 0973-1768,Number 1(2011), pp. 57-64.[5] Tarig M. Elzaki & Salih M. Elzaki, Application of New Transform "ElzakiTransform" to Partial Differential Equations, Global Journal of Pure and AppliedMathematics, ISSN 0973-1768,Number 1(2011), pp. 65-70.[6] Tarig M. Elzaki & Salih M. Elzaki, On the Connections Between Laplace andElzaki transforms, Advances in Theoretical and Applied Mathematics, ISSN 0973-4554 Volume 6, Number 1(2011),pp. 1-11.

[7] Tarig M. Elzaki & Salih M. Elzaki, On the Elzaki Transform and OrdinaryDifferential Equation With Variable Coefficients, Advances in Theoretical andApplied Mathematics. ISSN 0973-4554 Volume 6, Number 1(2011),pp. 13-18.

Received: August, 2011