Astrophysics

A new algorithm for solving di�erentialequations of Lane±Emden type

Abdul-Majid Wazwaz

Department of Mathematics and Computer Science, Saint Xavier University, 3700 West 103rd Street,

Chicago, IL 60655, USA

Abstract

In this paper, a reliable algorithm is employed to investigate the di�erential equations

of Lane±Emden type. The algorithm rests mainly on the Adomian decomposition

method with an alternate framework designed to overcome the di�culty of the singular

point. The proposed framework is applied to a generalization of Lane±Emden equations

so that it can be used in di�erential equations of the same type. Ó 2001 Elsevier Science

Inc. All rights reserved.

Keywords: Lane±Emden type equations; Chandrasekhar equation; Theory of stellar structure;

Adomian decomposition method

1. Introduction

Many problems in the literature of mathematical physics can be distinctivelyformulated as equations of Lane±Emden type [1,2] de®ned in the form

y 00 � 2

xy 0 � f �y� � 0; �1�

where f �y� is some given function of y. It is well known that an analytic so-lution of Lane±Emden type equation (1) is always possible [1] in the neigh-borhood of the singular point x � 0 for the boundary conditions

y�0� � y0; y 0�0� � 0: �2�

www.elsevier.com/locate/amcApplied Mathematics and Computation 118 (2001) 287±310

E-mail address: [email protected] (A.-M. Wazwaz).

0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved.

PII: S0 0 96 -3 0 03 (9 9 )0 02 2 3- 4

Eq. (1) with specializing f �y� was used to model several phenomena in math-ematical physics and astrophysics such as the theory of stellar structure, thethermal behavior of a spherical cloud of gas, isothermal gas spheres, andtheory of thermionic currents [1±3]. A substantial amount of work has beendone on this type of problems for various structures of f �y�.

In what follows, we present various forms of f �y� that have attracted theattention of the scienti®c community, due to their signi®cant applications.Among the most popular form of f �y� is

f �y� � ym; �3�

where m is a constant parameter. The well-known Lane±Emden equation ofindex m results from combining (1) and (3). The standard Lane±Emdenequation was used to model the thermal behavior of a spherical cloud of gasacting under the mutual attraction of its molecules [1] and subject to theclassical laws of thermodynamics.

Another interesting form of f �y� is given by

f �y� � �y2 ÿ C�3=2: �4�

Inserting (4) into (1) gives the ``white-dwarf '' equation introduced byChandrasekhar [2] in his study of the gravitational potential of the degeneratewhite-dwarf stars. It is interesting to point out that setting C � 0 reduces thewhite-dwarf equation to Lane±Emden equation of index m � 3.

Moreover, the other nonlinear forms of f �y� are the exponential functions

f �y� � e y�x� �5�and

f �y� � eÿy�x�: �6�

Substituting (5) into (1) results in a model that describes the isothermal gasspheres where the temperature remains constant and the index m is in®nite. Onthe other hand, combining (1) and (6) gives a model that appears in the theoryof thermionic currents thoroughly investigated by Richardson [3].

Furthermore, the function f �y� appears in eight additional cases [1], namely,four triangular forms de®ned by

f �y� � � sin y;� cos y; �7�

and four hyperbolic functions de®ned by

288 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310

f �y� � � sinh y;� cosh y; �8�A thorough discussion of the formulation of these models and the physicalstructure of the solutions can be found in [1±3].

A di�cult element in the analysis of this type of equations is the singularitybehavior that occurs at x � 0. The motivation for the analysis presented in thispaper comes actually from the purpose of introducing a reliable framework,mainly depends on Adomian decomposition method [4±6], to handle this typeof equations by reducing the amount of computations. Our next aim consists intesting the proposed algorithm in handling a generalization of this type ofproblems.

Most algorithms currently in use for handling the Lane±Emden typeproblems are based on either series solutions or perturbation techniques. Sofar, the series solution can be found by tedious computation when these al-gorithms are used.

In recent years a lot of attention has been devoted to the study of Adomiandecomposition method to investigate various scienti®c models. The Adomiandecomposition method which accurately computes the series solution is ofgreat interest to applied sciences. The method provides the solution in a rapidlyconvergent series with components that are elegantly computed. A more dif-®cult question to answer is how this method can be modi®ed to address theconcept of singular points. To properly address this question, we may requireslight variation of the decomposition algorithm as described [4±6]. An alternateframework can be designed to overcome the di�culty of the singular point atx � 0.

2. Analysis

In this section, we will introduce a reliable algorithm that consists of twodistinct approaches to handle in a realistic and e�cient way the Lane±Emdentype of problems. As will be discussed later, the proposed approaches dependmainly on Adomian decomposition method. In the ®rst approach, we apply thedecomposition method in a straightforward manner with a new choice for thedi�erential operator L. In the other approach, we will employ ®rst a well-de-®ned transformation used by Davis and Chandrasekhar [1,2] to remove theterm y 0 from Eq. (1). The Adomian decomposition method will be employed tothe resulting normal form.

In the following, we present an analysis of each approach.Approach I: It is well known that Adomian decomposition method usually

starts by de®ning the equation in an operator form by considering the highest-ordered derivative in the problem. However, a slight change is necessary toovercome the singularity behavior. The choice here is to de®ne the di�erential

A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310 289

operator L in terms of the two derivatives contained in the problem. Following[4,5], therefore we rewrite (1) in the form

L1y � ÿf �y�; �9�where the di�erential operator L1 employs, in order to overcome the singularityat x � 0, the ®rst two derivatives, hence de®ned by

L1 � xÿ2 d

dxx2 d

dx

� �: �10�

The inverse operator Lÿ11 is therefore considered a two-fold integral operator

de®ned by

Lÿ11 ��

Z x

0

xÿ2

Z x

0

x2�� dxdx: �11�

Operating with Lÿ11 on (9), it then follows

y�x� � aÿ Lÿ11 f �y�; �12�

where

a � y�0�; y 0�0� � 0: �13�The details of the Adomian decomposition analysis, that will be employed inthis approach, are presented elsewhere [4±8], but its application will be con-tained here. The Adomian decomposition method introduces the solution y�x�by a series of components

y�x� �X1n�0

yn�x�; �14�

and the nonlinear function f �y� by an in®nite series of polynomials

f �y� �X1n�0

An�y0; y1; . . . ; yn�; �15�

where the components yn�x� of the solution y�x� will be determined recurrently,and An are the Adomian polynomials that can be constructed for variousclasses of nonlinearity according to speci®c algorithms set of [4,5] and recentlycalculated by Wazwaz [9].

Substituting (14) and (15) into (12) givesX1n�0

yn�x� � aÿ Lÿ11

X1n�0

An�y0; y1; . . . ; yn�: �16�


Identifying y0�x� � a, the recursive relation

y0�x� � a;

yk�1�x� � ÿLÿ11 �Ak�; k P 0;

�17�

will lead to the complete determination of the components yn�x� of y�x�. Theseries solution of y�x� de®ned by (14) follows immediately.

The most striking advantage of using this choice for the operator L1 is that itattacks the problems of Lane±Emden type directly without any need for atransformation formula. In the above discussion it was shown that, with theproper choice of the di�erential operator L1, it is possible to overcome thesingularity question and to attain practically a series solution by computingcomponents of y�x� as far as we like. We will examine this approach moreclosely at a later point.

Approach II: In a manner parallel to [1,2,10], we ®rst remove the term y 0 in(1). This can be achieved by using the transformation

u�x� � xy�x� �18�so that

u 0 � xy 0 � y; u00 � xy 00 � 2y 0: �19�Substituting (18) and (19) into (1) gives

u00 � xfux

� �� 0: �20�

In view of (20), the di�culties encountered by the singularity at x � 0 in (1) iseliminated. Unlike the approach of Chandrasekhar [2], our next step consists ofapplying Adomian decomposition method to (20). The Adomian decomposi-tion method can be applied directly and in a straightforward manner to Eq.(20). The di�erential operator L2 that will be used here, considers only onederivative u00�x�. Based on this, we ®rst rewrite (20) in an operator form

L2u � ÿxfux

� �; �21�

where L2 is a second order di�erential operator, and Lÿ12 is a two-fold inte-

gration operator de®ned by

Lÿ12 ��

Z x

0

Z x

0

�� dxdx: �22�

Operating with Lÿ12 on both sides of (21) yields

u�x� � axÿ Lÿ12 xf

ux

� �� ; �23�


or equivalently

u�x� � axÿ Lÿ12 �xF �u��; �24�

where

u�0� � 0; a � u0�0�; �25�and

F �u� � fux

� �: �26�

The Adomian decomposition method de®nes the solution u�x� by a series ofcomponents

u�x� �X1n�0

un�x�; �27�

and the nonlinear function F �u� by a series of polynomials

F �u� �X1n�0

Bn�u0; u1; . . . ; un�; �28�

where the components un�x�; n P 0, will be determined recurrently with u0

identi®ed as ax, and Bn are the so-called Adomian polynomials that can begenerated according to speci®c algorithms as stated before. Substituting (27)and (28) into both sides of (24) gives

X1n�0

un�x� � axÿ Lÿ12 x

X1n�0

Bn�x� !

: �29�

The decomposition method admits the use of the recursive relation

u0�x� � ax;

uk�1�x� � ÿLÿ12 �xBk�x��; k P 0:

�30�

With u0 identi®ed in (30), this can be valuable in determining the other com-ponents. In view of (30), the function u�x�, and consequently the solution y�x�follows immediately in a series form upon using (27) and (18).

The approaches presented above can be validated by testing it on a varietyof several Lane±Emden type models. For a thorough discussion of formulationof these models, see [1,2]. For our purposes, it is su�cient to investigate thefollowing well known six physical models. Approach I will be applied to the®rst and the last two models. For comparison reasons, we will apply the twoapproaches to the other two models.


3. Applications

3.1. Lane±Emden equation of index m

The Lane±Emden equation of index m is a basic equation in the theory ofstellar structure [10]. The equation describes the temperature variation of aspherical gas cloud under the mutual attraction of its molecules and subject tothe laws of thermodynamics [1,2,10]. The Lane±Emden equation of index m isof the form

y 00 � 2

xy 0 � ym � 0; �31�

which has been the object of much study [1±4]. The boundary conditions,which are of most interest [1], are the following:

y�0� � 1; y 0�0� � 0: �32�It was physically shown that interesting values of m lie in the interval [0,5]. Inaddition, exact solutions exist only for m � 0; 1 and 5. For other values of m,series solutions are obtainable.

The general solution for Eq. (31) is to be constructed for all possible valuesof the index m;m P 0. Notice that Eq. (31) is linear for m � 0 and 1, andnonlinear otherwise. In an operator form, Eq. (31) becomes

L1y � ÿym: �33�Applying Lÿ1

1 to both sides of (33) and using the boundary conditions we ®nd

y�x� � 1ÿ Lÿ11 �ym�x��: �34�

Using the decomposition suggestions for y�x� and for the nonlinear term ym assuggested above by (14) and (15) we obtain

X1n�0

yn�x� � 1ÿ Lÿ11

X1n�0

An

!: �35�

The components yn�x� of the solution y�x� can be elegantly obtained by usingthe recursive relationship

y0�x� � 1;

yk�1�x� � ÿLÿ11 �Ak�; k � 0; 1; 2; . . . :

�36�

It is convenient to list the ®rst few Adomian polynomials. Following [5,9] wederive


A0 � ym0 ;

A1 � my1ymÿ10 ;

A2 � my2ymÿ10 � m�mÿ 1� y

21

2!ymÿ2

0 ;

A3 � my3ymÿ10 � m�mÿ 1�y1y2ymÿ2

0 � m�mÿ 1��mÿ 2� y31

3!ymÿ3

0 ;

A4 � my4ymÿ10 � m�mÿ 1� y2

2

2!

�� y1y3

�ymÿ2

0

� m�mÿ 1��mÿ 2� y21y2

2!ymÿ3

0

� m�mÿ 1��mÿ 2��mÿ 3� y41

4!ymÿ4

0 ;

�37�

and so on. Inserting (37) into (36) gives

y0 � 1;

y1 � ÿLÿ11 �A0� � ÿ 1

6x2;

y2 � ÿLÿ11 �A1� � m

120x4;

y3 � ÿLÿ11 �A2� � ÿm�8mÿ 5�

3 � 7!x6;

y4 � ÿLÿ11 �A3� � m�70ÿ 183m� 122m2�

9 � 9!x8;

y5 � ÿLÿ11 �A4� � m�3150ÿ 1080m� 12642m2 ÿ 5032m3�

45 � 11!x10:

�38�

This means that the solution in a series form is given by

y�x� � 1ÿ 1

6x2 � m

120x4 ÿ m�8mÿ 5�

3 � 7!x6

� m�70ÿ 183m� 122m2�9 � 9!

x8

� m�3150ÿ 1080m� 12642m2 ÿ 5032m3�45 � 11!

x10 � � � � ; �39�

which is in full agreement with the calculations of J.R. Airey [1]. Fig. 1 belowshows the Pad�e approximants [6/6] of y�x� in (39) for a few values of m. It iswell-known that Pad�e approximant has the advantage of manipulating apolynomial approximation into a rational function to gain more informationabout y�x�.

Substituting m � 0; 1 and 5 into (39) leads to the exact solution

y�x� � 1ÿ 1

3!x2; y�x� � sin x

x�40�


and

y�x� � 1

�� x2

3

�ÿ1=2

; �41�

respectively.

3.2. The white-dwarf equation

In this model we consider the ``white-dwarf '' equation

y 00 � 2

xy 0 � �y2 ÿ C�3=2 � 0; �42�

introduced by Davis [1] and Chandrasekhar [2] in his study of the gravitationalpotential of the degenerate white-dwarf stars. The boundary conditions of (42)are

y�0� � 1; y 0�0� � 0: �43�

It is clear that Eq. (42) is of Lane±Emden type where

f �y� � �y2 ÿ C�3=2: �44�

If C � 0, Eq. (42) reduces to Lane±Emden equation of index m � 3. For athorough discussion of the ``white-dwarf '' formula (42), see [2]. For our pur-poses, it is su�cient to handle the problem by using Approach I as introducedabove. Applying Lÿ1

1 to both sides of (42) yields

Fig. 1. Pade' approximants [6/6] of y�x� for few values of m.


y�x� � 1ÿ Lÿ11 �f �y��; �45�

where

f �y� � �y2 ÿ C�3=2: �46�

Following [4±6], the recursive relationship

y0�x� � 1;

yk�1�x� � ÿLÿ11 �Ak�; k P 0;

�47�

is readily obtained, where Ak are Adomian polynomials. It is convenient to listthe ®rst few Adomian polynomials constructed by

A0 � �y20 ÿ C�3=2

;

A1 � 3

2y1�y2

0 ÿ C�1=2;

A2 � 3

2y2�y2

0 ÿ C�1=2 � 3

8y2

1�y20 ÿ C�ÿ1=2

;

A3 � 3

2y3�y2

0 ÿ C�1=2 � 3

4y1y2�y2

0 ÿ C�ÿ1=2 ÿ 1

16y3

1�y0 ÿ C�ÿ3=2;

A4 � 3

2y4�y2

0 ÿ C�1=2 � 3

4

1

2!y2

2

�� y1y3

��y2

0 ÿ C�ÿ1=2

� 3

16y2

1y2�y20 ÿ C�ÿ3=2 � 3

128y4

1�y20 ÿ C�ÿ5=2

:

�48�

Substituting (48) into (47), the following components:

y0 � 1;

y1 � ÿLÿ11 �A0� � ÿ1=6�C ÿ 1�3=2x2;

y2 � ÿLÿ11 �A1� � 1

40�C ÿ 1�2x4;

y3 � ÿLÿ11 �A2� � ÿ 1

7!�5�C ÿ 1� � 14��C ÿ 1�5=2x6;

y4 � ÿLÿ11 �A3� � 1

3 � 9!�339�C ÿ 1� � 280��C ÿ 1�3x8;

y5 � ÿLÿ11 �A4�

� 1

5 � 11!�1425�C ÿ 1�2 � 11436�C ÿ 1� � 4256��C ÿ 1�7=2x10;

�49�


are readily computed. The series solution is given by

y�x� � 1ÿ 1

6q3x2 � 1

40q4x4 ÿ q5

7!�5q2 � 14�x6 � q6

3 � 9!�339q2 � 280�x8

� q7

5 � 11!�1425q4 � 11436q2 � 4256�x10 � � � � ; �50�

where q2 � C ÿ 1. Fig. 2 shows the Pad�e approximants [6/6] of y�x� for fewvalues of C. As indicated before, Pad�e approximant has the advantage ofmanipulating the polynomial approximation into a rational function to gainmore information about y�x�.

3.3. Isothermal gas spheres equation

In the following, we consider the di�erential equation of isothermal gasspheres

y 00 � 2

xy 0 � e y � 0; �51�

subject to the boundary conditions

y�0� � y 0�0� � 0: �52�This model can be used to view the isothermal gas spheres, where the tem-perature remains constant and the index m is in®nite. For a thorough discus-sion of the formulation of (51), see [1].

It is interesting to note that

y � ln2

x2

� �; �53�

Fig. 2. Pade' approximants [6/6] of y�x� for few values of C.


is the only known particular solution [1]. However, for comparison reasons,this model will be handled by applying the two approaches presented above.

Approach I: In an operator form, Eq. (51) is written in the form

L1y � ÿe y : �54�Operating with Lÿ1

1 on both sides of (54) we ®nd

y � ÿLÿ11 �e y�: �55�

Using the decomposition series for the linear function y�x� and the polynomialseries for the nonlinear term and proceeding as before we obtain the recursiverelationship

y0�x� � 0;

yk�1�x� � ÿLÿ11 �Ak�; k P 0:

�56�

The Adomian polynomials for the nonlinear term eÿy are computed as follows:

A0 � e y0 ;

A1 � y1e y0 ;

A2 � y2

�� y2

1

2!

�e y0 ;

A3 � y3

�� y1y2 � y3

1

3!

�e y0 ;

A4 � y4

�� y1y3 � y2

2

2!� 1

2y2

1y2 � 1

4y4

1

�e y0 ;

�57�

obtained by using formal algorithms [5,9]. Substituting (57) into (56) gives thecomponents

y0 � 0;

y1 � ÿLÿ11 �A0� � ÿ 1

6x2;

y2 � ÿLÿ11 �A1� � 1

5 � 4!x4;

y3 � ÿLÿ11 �A2� � ÿ 8

21 � 6!x6;

y4 � ÿLÿ11 �A3� � 122

81 � 8!x8;

y5 � ÿLÿ11 �A4� � ÿ 4087

495 � 10!x10;

�58�


so that other components can be evaluated in a like manner. In view of (58), thesolution in a series form is given by

y�x� � ÿ 1

6x2 � 1

5 � 4!x4 ÿ 8

21 � 6!x6 � 122

81 � 8!x8 ÿ 61 � 67

495 � 10!x10 � � � � �59�

Approach II: In this approach we use the substitution u � xy, hence theequation can be rewritten in the form

L2u � ÿxeu=x; �60�where L2 is a second order di�erential operator. Operating with the inverseoperator Lÿ1

2 and using the decomposition assumptions for linear and non-linear terms gives the recursive relationship

u0�x� � 0;

uk�1�x� � ÿLÿ12 �xBk�; k P 0;

�61�

where Bk are the Adomian polynomials that represent the nonlinear term eu=x.Following [5,9], the Adomian polynomials can be shown to be as

B0 � eu0=x;

B1 � 1

xu1eu0=x;

B2 � 1

xu2

�� 1

2! � x2u2

1

�eu0=x;

B3 � 1

xu3

�� 1

x2u1u2 � 1

3! � x3u3

1

�eu0=x:

�62�

Substituting (62) into (61) and proceeding as before we obtain

u0 � 0;

u1 � ÿLÿ12 �xB0� � ÿ 1

6x3;

u2 � ÿLÿ12 �xB1� � 1

120x5;

u3 � ÿLÿ12 �xB2� � ÿ 8

21 � 6!x7;

u5 � ÿLÿ12 �xB3� � 122

81 � 8!x9;

�63�

such that other components can be computed in a like manner to enhance theaccuracy level. In view of (63), the series solution (59) follows immediatelyupon noting that y�x� � u�x�=x.


3.4. Richardson's theory of thermionic currents

Richardson [3] introduced a counterpart of Eq. (51) in which ey�x� is replacedby eÿy�x�. The model is controlled by the nonlinear di�erential equation

y 00 � 2

xy 0 � eÿy � 0; �64�


y�0� � y 0�0� � 0: �65�This model appears in Richardson's theory of thermionic currents when thedensity and electric force of an electron gas in the neighborhood of a hot bodyin thermal equilibrium [1] is to be determined. For a thorough discussion of theformulation of (64) and the physical behavior of the emission of electricityfrom hot bodies, see [1,3].

Following the discussion of the previous application, this model can behandled by applying the two approaches presented above.

Approach I: Operating with Lÿ11 on both sides of (64) leads to

y � ÿLÿ11 �eÿy�: �66�

Using the decomposition series assumptions gives the recursive relationship

y0�x� � 0;

yk�1�x� � ÿLÿ11 �Ak�; k P 0:

�67�

The Adomian polynomials for the nonlinear term eÿy are computed as follows:

A0 � eÿy0 ;

A1 � ÿy1eÿy0 ;

A2 ��ÿ y2 � y2

1

2!

�eÿy0 ;

A3 ��ÿ y3 � y1y2 ÿ y3

1

3!

�eÿy0 ;

A4 ��ÿ y4 � y1y3 � y2

2

2!ÿ y2

1y2

2!� y4

1

4!

�eÿy0 :

�68�


Substituting (68) into (67) gives the components

y0 � 0;

y1 � ÿLÿ11 �A0� � ÿ 1

6x2;

y2 � ÿLÿ11 �A1� � ÿ 1

5 � 4!x4;

y3 � ÿLÿ11 �A2� � ÿ 8

21 � 6!x6;

y4 � ÿLÿ11 �A3� � ÿ 122

81 � 8!x8;

y5 � ÿLÿ11 �A4� � ÿ 61 � 67

495 � 10!x8;

�69�

where other components can be evaluated in a like manner. In view of (69), thesolution in a series form is given by

y�x� � ÿ 1

6x2 ÿ 1

5 � 4!x4 ÿ 8

21 � 6!x6 ÿ 122

81 � 8!x8 ÿ 61 � 67

495 � 10!x8 � � � � �70�

Approach II: As discussed before, we apply the inverse operator Lÿ12 and use

the decomposition assumptions for linear and nonlinear terms to obtain therecursive relationship

u0�x� � 0;

uk�1�x� � ÿLÿ12 �xBk�; k P 0;

�71�

where Bk are Adomian polynomials that represent the nonlinear term eÿu=x.Following [5,9], Adomian polynomials can be derived as follows:

B0 � eÿu0=x;

B1 � ÿ 1

xu1eÿu0=x;

B2 ��ÿ 1

xu2 � 1

2! � x2u2

1

�eÿu0=x;

B3 ��ÿ 1

xu3 � 1

x2u1u2 ÿ 1

3! � x3u3

1

�eÿu0=x:

�72�


Substituting (72) into (71) and proceeding as before we obtain

u0 � 0;

u1 � ÿLÿ12 xB0 � ÿ 1

6x3;

u2 � ÿLÿ12 xB1 � ÿ 1

120x5;

u3 � ÿLÿ12 xB2 � ÿ 1

1890x7;

u4 � ÿLÿ12 xB3 � 122

81 � 8!x9;

�73�

and so on. In view of (73), the series solution (70) follows immediately uponnoting that y�x� � u�x�=x.

It is interesting to point out that eight possible cases are discussed in [1] inwhich f �y� is specialized by

f �y� � � sin y;� cos y;� sinh y;� cosh y: �74�

In the following, Approach I will be applied to one trigonometric function andto one hyperbolic function. Other cases can be handled in a like manner.

3.5. Lane±Emden type where f �y� � sin y

Consider the di�erential equation

y 00 � 2

xy 0 � sin y � 0; �75�


y�0� � 1; y 0�0� � 0: �76�Proceeding as before we obtain the relation

y0�x� � 1;

yk�1�x� � ÿLÿ11 �Ak�; k P 0;

�77�

where Ak are Adomian polynomials that represent the nonlinear termsin y. Following Adomian algorithm [5], we list the ®rst few polynomials givenby


A0 � sin y0;

A1 � y1 cos y0;

A2 � y2 cos y0 ÿ y21

2!sin y0;

A3 � y3 cos y0 ÿ y1y2 sin y0 ÿ y31

3!cos y0;

A4 � y4

�ÿ 1

2y2

1y2

�cos y0 ÿ 1

2y2

2

�� y1y3 ÿ 1

4!y4

1

�sin y0;

�78�

so that other polynomials can be generated as well. Substituting (78) into (77)gives the ®rst components of y�x� by

y0�x� � 1;

y1�x� � ÿLÿ1�A0� � ÿ 1

6k1x2;

y2�x� � ÿLÿ11 �A1� � 1

120k1k2x4;

y3�x� � ÿLÿ11 �A2� � k1

1

3024k2

1

�ÿ 1

5040k2

2

�x6;

y4�x� � ÿLÿ11 �A3� � k1k2

�ÿ 113

3265920k2

1 �1

362880k2

2

�x8;

y5�x� � ÿLÿ11 �A4�

� k1

1781

898128000k2

1k22

�ÿ 1

39916800k4

2 ÿ19

23950080k4

1

�x10;

�79�

where

k1 � sin 1; k2 � cos 1: �80�This in turn gives

y�x� � 1ÿ 1

6k1x2 � 1

120k1k2x4 � k1

1

3024k2

1

�ÿ 1

5040k2

2

�x6

� k1k2

�ÿ 113

3265920k2

1 �1

362880k2

2

�x8

� k1

1781

898128000k2

1k22

�ÿ 1

39916800k4

2 ÿ19

23950080k4

1

�x10 �O�x11�:

�81�


On the other hand, in [1], the cos y model

y 00 � 2

xy 0 � cos y � 0; y�0� � 1� p=2; y 0�0� � 0 �82�

was investigated. Proceeding as before in the sin y model, we obtain the seriessolution for (82) de®ned by

y�x� � 1�p=2� 1

6k1x2� 1

120k1k2x4� k1

�ÿ 1

3024k2

1 �1

5040k2

2

�x6

� k1k2

�ÿ 113

3265920k2

1 �1

362880k2

2

�x8

� k1

�ÿ 1781

898128000k2

1k22 �

1

39916800k4

2 �19

23950080k4

1

�x10�O�x11�:

�83�In a like manner we can derive the series solution for the ÿ sin y and ÿ cos ymodels.

In the following, we investigate the sinh y model.

3.6. Lane±Emden type where f �y� � sinh y

Consider the di�erential equation

y 00 � 2

xy 0 � sinh y � 0; �84�


y�0� � 1; y 0�0� � 0: �85�Following the above discussions, Eq. (84) can be transformed to the relation

y0�x� � 1;

yk�1�x� � ÿLÿ11 �Ak�; k P 0;

�86�

where Ak are Adomian polynomials that represent the nonlinear term sinh y.The ®rst few polynomials can be generated as follows:

A0 � sinh y0;

A1 � y1 cosh y0;

A2 � y2 cosh y0 � y21

2!sinh y0;

A3 � y3 cosh y0 � y1y2 sinh y0 � y31

3!cosh y0;

�87�

so that other polynomials can be generated as well. Substituting (87) into (86)gives the ®rst components of y�x� by


y0�x� � 1;

y1�x� � ÿLÿ11 �A0� � ÿ e2 ÿ 1

12ex2;

y2�x� � ÿLÿ11 �A1� � e4 ÿ 1

480e2x4;

y3�x� � ÿLÿ11 �A2� � ÿ 2e6 � 3e2 ÿ 3e4 ÿ 2

30240e3x6;

y4�x� � ÿLÿ11 �A3� � 61e8 ÿ 104e6 � 104e2 ÿ 61

26127360e4x8:

�88�

The series solution is therefore

y�x� � 1ÿ e2 ÿ 1

12ex2 � e4 ÿ 1

480e2x4 ÿ 2e6 � 3e2 ÿ 3e4 ÿ 2

30240e3x6

� 61e8 ÿ 104e6 � 104e2 ÿ 61

26127360e4x8 �O�x9�: �89�

We can easily show that, the cosh y model [1]

y 00 � 2

xy 0 � cosh y � 0; y�0� � 1; y 0�0� � 0; �90�

gives the series solution

y�x� � 1ÿ e2 � 1

12ex2 � e4 ÿ 1

480e2x4 ÿ 2e6 � 3e2 � 3e4 � 2

30240e3x6

� 61e8 � 104e6 ÿ 104e2 ÿ 61

26127360e4x8 �O�x9�: �91�

The other models, namely, the ÿ sinh y and ÿ cosh y can be handled in a similarfashion.

4. Generalization

In this section we will examine more closely Lane±Emden like equations.The standard coe�cient of y 0 in Lane±Emden equation is 2=x. However, if wereplace 2=x by n=x, for real n; n P 0, then we write down Lane±Emden likeequation in a general fashion as

y 00 � nx

y 0 � f �y� � 0; n P 0; �92�

with boundary conditions given by

y�0� � a; y 0�0� � 0: �93�


It is convenient to consider a modi®cation to Approach I in order to enable usto handle Eq. (92) for all real values of n P 0. Recall that for n � 0, theboundary x � 0 is an ordinary point, whereas for n 6� 0 the boundary x � 0 is asingular point. This makes our discussion of the general model (92) valuableand promising to handle a huge size of di�erential equations of the same typewith and without a singular point.

Our strategy begins by introducing the di�erential operator

L � xÿn d

dxxn d

dx

� �; �94�

for which the inverse operator Lÿ1 is expressed by

Lÿ1�� Z x

0

xÿn

Z x

0

xn��dxdx: �95�

In an operator form, Eq. (92) may be rewritten as

Ly � ÿf �y�: �96�

Operating with Lÿ1 carries (96) into

y � aÿ Lÿ1�f �y��: �97�

The slight change we imposed in de®ning the operator L in (94), in terms ofthe ®rst two derivatives, was successful to overcome the singularity issue forn 6� 0.

As discussed above, Adomian decomposition method introduces the de-composition series

y�x� �X1n�0

yn�x�; �98�

and the in®nite series of polynomials

f �y� �X1n�0

An�y0; y1; . . . ; yn�; �99�

where the components yn�x� of the solution y�x� will be determined recurrently,and An are Adomian polynomials.

Substituting (98) and (99) into (97) gives

X1n�0

yn�x� � aÿ Lÿ1X1n�0

An�y0; y1; . . . ; yn�: �100�


Identifying y0�x� � a, the recursive relation

y0�x� � a;

yk�1�x� � ÿLÿ1�Ak�; k P 0;�101�

or equivalently

y0�x� � a;

yk�1�x� � ÿZ x

0

xÿn

Z x

0

xn�Ak�dxdx; k P 0;�102�

will lead to the complete determination of the components yn�x� of y�x�. Theseries solution of y�x� follows immediately.

In the following, we will examine this approach more closely by consideringf �y� � ym;m P 0. Other forms of f �y� can be handled in a similar way.

For our purposes, we consider

y 00 � nx

y 0 � ym � 0; �103�

where the boundary conditions, which are of most interest are the following:

y�0� � 1; y 0�0� � 0: �104�Using the recursive relation (102) yields

y0�x� � a;

yk�1�x� � ÿZ x

0

xÿn

Z x

0

xn�Ak�dxdx; k P 0:�105�

The ®rst few Adomian polynomials for f �y� � ym were calculated beforeand given by (37). Accordingly, the ®rst few components of y�x� are givenby

y0 � 1;

y1 � ÿLÿ1�A0� � ÿZ x

0

xÿn

Z x

0

xn dxdx � ÿ 1

2�n� 1� x2

y2 � ÿZ x

0

xÿn

Z x

0

xnA1 dxdx

� m2�n� 1�

Z x

0

xÿn

Z x

0

xnx2 dxdx

� m8�n� 1��n� 3� x

4;


y3 � ÿZ x

0

xÿn

Z x

0

xnA2 dxdx

� ÿ�2n� 4�m2 ÿ �n� 3�m8�n� 1�2�n� 3�

Z x

0

xÿn

Z x

0

xnx4 dxdx

� ÿ �2n� 4�m2 ÿ �n� 3�m48�n� 1�2�n� 3��n� 5� x

6;

y4 � ÿZ x

0

xÿn

Z x

0

xnA3 dxdx;

� �6n2 � 32n� 34�m3 ÿ �7n2 � 46n� 63�m2 � �2n2 � 16n� 30�m48�n� 1�3�n� 3��n� 5�

�Z x

0

xÿn

Z x

0

xnx6 dxdx;

� �6n2 � 32n� 34�m3 ÿ �7n2 � 46n� 63�m2 � �2n2 � 16n� 30�m384�n� 1�3�n� 3��n� 5��n� 7� x8:

�106�Consequently, the series solution for all real values of n P 0 and m P 0 is ex-pressed in the form

y�x� � 1ÿ 1

2�n� 1� x2 � m

8�n� 1��n� 3� x4 ÿ �2n� 4�m2 ÿ �n� 3�m

48�n� 1�2�n� 3��n� 5� x6

� �6n2 � 32n� 34�m3 ÿ �7n2 � 46n� 63�m2 � �2n2 � 16n� 30�m384�n� 1�3�n� 3��n� 5��n� 7� x8 � � � �

�107�As indicated before, the general solution obtained in (107) works for all realvalues of n and m. For example, for ®xed n � 0, the following solutions

y�x� � 1ÿ 1

2x2;

y�x� � cos x;

y�x� � 1ÿ 1

2x2 � 1

12x4 ÿ 1

72x6 � 1

504x8 � � � � ;

�108�

can be obtained for m � 0; 1 and 2 respectively upon using (107), where x� 0 isan ordinary point. In addition, for ®xed n� 0.5, we ®nd


y�x� � 1ÿ 1

3x2;

y�x� � 1ÿ 1

3x2 � 1

42x4 ÿ 1

1386x6 � 1

83160x8 � � � � ;

y�x� � 1ÿ 1

3x2 � 1

21x4 ÿ 13

2079x6 � 23

31185x8 � � � � ;

�109�

obtained for m � 0; 1 and 2 respectively, where x� 0 is a singular point.Moreover, the solutions

y�x� � 1ÿ 1

4x2;

y�x� � 1ÿ 1

4x2 � 1

64x4 ÿ 1

2304x6 � 1

147456x8 � � � � ;

y�x� � 1ÿ 1

4x2 � 1

32x4 ÿ 1

288x6 � 13

36864x8 � � � � ;

�110�

obtained for n� 1 and m� 0, 1, and 2 respectively.

5. Conclusion

In the above discussion it was shown that, with the proper investment ofAdomian decomposition method, it is possible to attain an analytic solution toLane±Emden type of equations. The di�culty in using Adomian decomposi-tion method directly to this type of equations, due to the existence of singularpoint at x� 0, is overcome here.

Our goal has been achieved by formally deriving analytical approximationswith a high degree of accuracy. The computational size has been reduced andthe rapid convergence has been guaranteed [11].

Lane±Emden like equation was generalized, by changing the coe�cient ofy 0, and the proposed technique was presented in a general way so that it can beused in applied sciences for similar cases.

In closing, we point out that other types of ODEs with singular coe�cients,such as Legender's equation, Laguerre's equation, and Bessel's equation, werehandled di�erently, but successively, by Adomian [12].

References

[1] H.T. Davis, Introduction to Nonlinear Di�erential and Integral Equations, Dover, New York,

1962.

[2] S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover, New York, 1967.

[3] O.U. Richardson, The Emission of Electricity from Hot Bodies, London, 1921.


[4] G. Adomian, R. Rach, N.T. Shawagfeh, On the analytic solution of Lane±Emden equation,

Foundations of Phys. Lett. 8 (2) (1995) 161±181.

[5] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer,

Boston, 1994.

[6] A.M. Wazwaz, A First Course in Integral Equations, World Scienti®c, River Edge, NJ, 1997.

[7] A.M. Wazwaz, A reliable modi®cation of Adomian's decomposition method, Appl. Math.

Comput. 102 (1999) 77±86.

[8] A.M. Wazwaz, Analytical approximations and Pade's approximants for Volterra's population

model, Appl. Math. Comput. 100 (1999) 13±25.

[9] A.M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear

operators, Appl. Math. Comput. 111 (2000) 33±51.

[10] N.T. Shawagfeh, Nonperturbative approximate solution for Lane±Emden equation, J. Math.

Phys. 34 (9) (1993) 4364±4369.

[11] Y. Cherrault, Convergence of Adomian's method, Kybernotes 18 (2) (1989) 31±38.

[12] G. Adomian, Di�erential coe�cients with singular coe�cients, Appl. Math. Comput. 47

(1992) 179±184.


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