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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�
290 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310
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�
A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310 291
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.
292 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310
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
A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310 293
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�
294 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310
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.
A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310 295
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�
296 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310
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.
A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310 297
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�
298 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310
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.
A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310 299
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�
subject to the boundary conditions
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�
300 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310
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�
A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310 301
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�
subject to the boundary conditions
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
302 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310
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�
A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310 303
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�
subject to the boundary conditions
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
304 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310
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�
A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310 305
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�
306 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310
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;
A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310 307
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
308 A.-M. Wazwaz / Appl. Math. Comput. 118 (2001) 287±310
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].
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