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International Electronic Journal of Pure and Applied MathematicsVolume 1 No. 1 2010, 37-45

SERIES EXPANSION AND MODIFIED DECOMPOSITION

METHODS FOR LANE-EMDEN EQUATIONS OF INDEX k

M. Nili Ahmadabadi1, F.M. Maalek Ghaini2

1,2Department of MathematicsUniversity of Yazd

Yazd, 89195-741, IRAN1e-mail: [email protected]

2e-mail: [email protected]

Abstract: In this study, Lane-Emden equations of index k are treated using seriesexpansion and modified Adomian decomposition methods. The special structureof this equation has been exploited to obtain two numerically efficient algorithmssuitable for computer programming. We will show that the Taylor series of theanswer can be found by direct substitution. We will also show that we do not needto compute Adomians polynomials for these equations.

AMS Subject Classification: 34A30, 65L05Key Words: Lane-Emden equations of index k, series expansion method, Adomiandecomposition method, Adomian polynomials

1. Introduction

The Lane-Emden equation of index k is a basic equation in the theory of stellar

structure, see [13]. The equation describes the temperature variation of a sphericalgas cloud under the mutual attraction of its molecules and subject to the laws ofthermodynamics [13], [9], [7]. The Lane-Emden equation of index k is of the form

y +2

xy + yk = 0, (1)

which has been the object of much study [9] [7] [12] [5] The boundary conditionstronicJournalofPureandAppliedMathematicsIEJPAM,

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38 M. Nili Ahmadabadi, F.M. Maalek Ghaini

At the beginning of 1980s Adomian proposed a new method to solve somefunctional equations [2], [3]. This method and its modifications [10], [14], [15], [16]have been efficiently used to solve singular and nonsingular ODEs. The theoriticaltreatment of the convergence of ADM has been considered in [1], [4], [6], [8].

In this paper, we are going to utilize series expansion and modified decomposition

methods in efficient ways for solving Lane-Emden equations of index k. The majordifficulty in using ADM is computing Adomians polynomials. Here we introduce ascheme which does not need to compute Adomians polynomials. We will also showthat the Taylor series of the answer can be found by direct substitution.

2. Analysis of the Methods

Suppose H is a Hilbert space and consider the following functional equation:

y = f + N y, (2)

where N : H H is a nonlinear operator on H, f is a given function in H and weare looking for y H satisfying (2).

At the beginning of 1980s Adomian developed a very powerful technique forsolving (2) where the solution y is considered in the form of the series:

y =i=1

ui (3)

and N y is expanded in the form of Adomian series:

N y =

n=0

An. (4)

The method consists of the following scheme:u0 = f,un+1 = An(u0, u1,...,un),

(5)

where each An is a polynomial in u0, u1,...,un called an Adomian polynomial ob-tronicJournalofPureandAppliedMathematicsIEJPAM,

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SERIES EXPANSION AND MODIFIED DECOMPOSITION... 39

Consider the initial value problem of Lane-Emden equation formulated as

y

+ 2x

y

+ F(x, y) = g(x), 0 < x 1,

y(0) = A, y

(0) = B,(8)

where A and B are given constants, F(x, y) is a given real function and g(x) C

[0, 1]is given. Usually, the standard ADM may be divergent in solving singular Lane-Emden equations. To overcome the singularity behavior, Wazwaz [15] defined thedifferential operator L in terms of two derivatives contained in the problem. Hewrote (8) in the form

L(y) = F(x, y) + g(x), (9)

where the differential operator L is defined by

L = x2d

dx(x2

d

dx). (10)

We must mention that even when both schemes are convergent, the scheme proposedby Waswas is usually faster [11].

2.1. Modified Decomposition Method

If we take L = x2 ddx(x2 ddx), equation (1) becomes

Ly = yk. (11)

Now we set y =

n=0 un and yk =

n=0 An and in an obvious way by putting

u0 = y0,un = x0 x2 x0 x2An1dxdx, n = 1, 2, . . . . (12)

we can find uns. To obtain Ans we exploit the following fact: In the expansion of(u0+u1+u2+...)

k we have terms in the general form k!k1!k2!...kl!

uk1i1 uk2i2

...uklil

, where kisare non-negative integers satisfying k1 + k2 + ... + kl = k. Since A0 should dependonly on u0, the only term that appears in A0 is u

k0 whose coefficient is

k!k! = 1.

Now u1 comes into account and the terms appearing in A1 are uk10 u1, u

k20 u

21,...,t

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2.2. Series Expansion Method

If we want to substitute y =

n=0 anxn in y

+ 2x

y

+ yk = 0 and compute ais,the only thing we need is to expand yk = (a0 + a1x + a2x

2 + ...)k and order itby powers of x obtaining yk = b0 + b1x + b2x

2 + ..., where bis depend on ais.

Again we exploit the fact that in the expansion of (u0 + u1 + u2 + ...)k

we haveterms in the general form k!

k1!k2!...kl!uk1i1 u

k2i2

...uklil

, where kis are non-negative integers

satisfying k1 + k2 + ... + kl = k. So we have yk = (u0 + u1 + u2 + ...)

k, where u0 = a0,u1 = a1x, u2 = a2x

2, ... The only term containing x0 is uk0 whose coefficientis k!

k! so we have b0 = ak0. The term containing x

1 is uk10 u1 whose coefficient isk!

1!(k1)!so we have b1 = ka

k10 a1. The terms containing x

2 are uk10 u2 and uk20 u

21

(assuming k 2) with coefficients k!(k1)!1! andk!

(k2)!2! , respectively. So we have

b2 = kak10 a2 +

k(k1)2 a

k20 a21 and so on. Note that even if k = 1 and we assume

the existence of uk20 u21, then its coefficient automatically becomes zero and we will

obtain b2 = a2. This fact simplifies the computer programming.

3. Examples

In this section we give a couple of examples to demonstrate the power of the proposed

algorithms.

Example 1. Consider the initial value problemy

+2

xy + y2 = 0,

y(0) = 1.(14)

Proposed Modified Decomposition Method. Implementing the method

described in Section 2.1. we obtain

(u0 + u1 + u2 + ...)2 = A0 + A1 + A2 + ...,

whereA0 = u

20,

A 2 + 2tr

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implyA0 = 1

u1 = 1

6x2,

A1 =

1

3 x

2

+

1

36 x

4

,

u2 =1

60x4

1

1512x6,

A2 =1

30x4

13

1890x6 +

113

226800x8

1

45360x10 +

1

2286144x12,

u3 = 1

1260x6 +

13

136080x8

113

24948000x10 +

1

7076160x12

1

480090240x14.

Accepting y = u0 + u1 + u2 + u3 as the approximate solution and substituting it inN[y] = y + 2

xy + y2, we obtain the local truncation error to be

N[y] = .001587301587x6 + o(x8)

which is of order x6. Note that we have done only three iterations.

Series Expansion Method. We assume the answer to be of the form y =a0+a1x+a2x

2+.... So we have y = a1+2a2x+3a3x2+..., y = 2a2+6x+12a4x

2+...

and implementing the method described in Section 2.2. we obtain

y2 = (a0 + a1x + a2x2 + ...)2 = a20 + 2a0a1x + (a

21 + 2a0a2)x

2 + ... .

Substituting in (14) we have

2a2+6a3x+12a4x2+

2a1x

+4a2+6a3x+8a4x2+a20+2a0a1x+(a

21+2a0a2)x

2+... = 0

and by putting the coefficients of xi equal to zero we obtain

a1 = 0,

a2 = a206

,

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Using the initial condition in (14), i.e. y(0) = 1, we obtain a0 = 1. Therefore weobtain the Taylor polynomial of the solution to be

y = 1 1

6x2 +

1

60x4 + ....

Note that the obtained three terms and the first three terms we obtained usingproposed modified decomposition method are the same. If we substitute y = 1 +16x

2 + 160x4 in N(y) = y + 2

xy + y2, we obtain the local truncation error to be

N(y) = 11

180x4

1

180x6 +

1

3600x8.

Example 2. Consider the initial value problem

y

+ 2x

y + y7 = 0,

y(0) = 1.(15)

Proposed Modified Decomposition Method. Implementing the methoddescribed in Section 2.1. we obtain

(u0 + u1 + u2 + ...)7 = A0 + A1 + ...,

whereA0 = u

70,

A1 = u71 + 7u0u

61 + 21u

20u

51 + 35u

30u

41 + 35u

40u

31 + 21u

50u

21 + 7u

60u1.

The above equations together with the recursive relations:u0 = 1,un =

x

0 x2

x

0 x2An1dxdx, n N,

implyA0 = 1,

u1 =1

6x2,

A7 2 7 4 35 6 35 8 7 10 1 14t

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which is of order x4. Note that we have done only two iterations.

Series Expansion Method. We assume the answer to be of the form y =a0+a1x+a2x

2+.... So we have y = a1+2a2x+3a3x2+..., y = 2a2+6x+12a4x

2+...and implementing the method described in Section 2.2. we obtain

y7 = (a0 + a1x + a2x2 + ...)7 = a70 + 7a60a1x + (21a50a21 + 7a60a2)x2 + ... .

Substituting in (15) we have

2a2+6a3x+12a4x2+

2a1x

+4a2+6a3x+8a4x2+a70+7a

60a1x+(21a

50a

21+7a

60a2)x

2+... = 0

and by putting the coefficients of xi equal to zero we obtain

a1 = 0,

a2 = a706

,

a3 = 0,

a4 = 7a130

120,

...

So we have

y = a0 a706

x2 +7a130120

x4 + ... .

Using the initial condition in (15), i.e. y(0) = 1, we obtain a0 = 1. Therefore weobtain the Taylor series of the solution to be

y = 1 +1

6x2

7

120x4 + ....



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4. Conclusion

Despite the fact that Lane-Emden equations of index k are non-linear, Taylor se-ries of their solution can be found by direct substitution. In order to do so, weexploited these equations special structure. We also exploited this special structure

in order to avoid computing Adomian polynomials while implementing a modifieddecomposition method.

References

[1] K. Abbaoui, Y. Cherruault, New ideas for proving convergence of decopmositionmethods, Comput. Math. Appl., 29 (1995), 103-108.

[2] G. Adomian, A review of the decomposition method and some recent resultsfor nonlinear equation, Math. Comput. Model., 13, No. 7 (1992), 17.

[3] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method,Kluwer, Boston, MA (1994).

[4] G. Adomian, R. Rach, Noise terms in decomposition series solution, Comput.Math. Appl., 24, No. 11 (1992), 61.

[5] G. Adomian, R. Rach, N.T. Shawagfeh, On the analytic solution of Lane-Emdenequation, Foundations of Phys. Lett., 8, No. 2 (1995), 161-181.

[6] B. Babolian, J. Biazar, On the order of convergence of Adomian method, Appl.Math. Comput., 130 (2002), 383-387.

[7] S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover, New

York (1967).

[8] Y. Cherruault, G. Adomian, K. Abbaoui, R. Rach, Further remarks on conver-gence of decomposition method, Bio-Med. Comput., 38 (1995), 89-93.

[9] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations,D N Y k (1962)


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[13] N.T. Shawagfeh, Nonperturbative approximate solution for Lane-Emden equa-tion, J. Math. Phys., 34, No. 9 (1993), 4364-4369.

[14] A.M. Wazwaz, A reliable modification of Adomian decomposition method,Appl. Math. Comput., 102 (1999), 77-86.

[15] A.M. Wazwaz, A new methd for solving singular initial value problems in thesecond-order ordinary diferential equations, Appl. Math. Comput., 128 (2002),45-57.

[16] A.M. Wazwaz, S.M. El-Sayed, A new modification of Adomian decompositionmethod for liear and nonlinear operators, App. Math. Comput., 122 (2001),393-405.


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