Solution of Lane-Emden Type Equations by Combination of the Spectral Method and Adomian Decomposition Method

Abstract

The application of a new modified Adomian decomposition method for obtaining the analytic solution of Lane-Emden type equations is investigated. The proposed method, called the spectral Adomian decomposition method, is based on a combination of spectral method and Adomian decomposition method. A comparative study between the proposed method and Adomian decomposition method is presented. The obtained result reveals that method is of higher efficiency, validity, and accuracy.

Academic Editor:Maria Gandarias

Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Received 26 December 2014; Revised 1 April 2015; Accepted 6 April 2015; 28 April 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Recently, the studies on singular initial value problems (IVPs) for second order ordinary differential equations (ODEs) have been the focus of considerable attention. One of the second order equations describing this type of problem is the Lane-Emden singular IVPs, which can be written in the form of [figure omitted; refer to PDF] subject to initial conditions: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are constants, [figure omitted; refer to PDF] is a continuous real valued function, and [figure omitted; refer to PDF] . The Lane-Emden equation was first studied by two astrophysicists, Jonathan Homer Lane and Robert Emden, who examined the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules subject to the classical laws of thermodynamics [1]. The well-known Lane-Emden equation has been applied to model several phenomena in mathematical physics and astrophysics such as theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres, and the theory of thermionic currents [2]. Approximate solutions to the abovementioned problems were presented by Wazwaz [3, 4] by applying the Adomian decomposition method which provides a convergent series solution. Nouh [5] accelerated the convergence of a power series solution of the Lane-Emden equation by using an Euler-Able transformation and pade approximation. Exact solution of generalized Lane-Emden solutions of the first kind is investigated by Goenner and Havas [6]. Liao [7] solved Lane-Emden type equations by applying a homotopy analysis method. He [8] obtained an approximate analytical solution of the Lane-Emden equation by applying a variational approach which uses a semi-inverse method. Yousefi [9] converted the Lane-Emden equation to an integral equation and then used Legendre wavelets and achieved an approximate solution for [figure omitted; refer to PDF] . Momoniat and Harley [10] applied the Lie Group method successfully to the generalized Lane-Emden equation of the first kind. The solution of a class of singular second order IVPs of Lane-Emden type was presented by Yildirim and Özis [11] and also Chowdhury and Hashim [12] by He's homotopy perturbation method. Ramos [13] provided a series approach to the Lane-Emden equation and compared it with He's homotopy perturbation method. The exact solutions of this equation (for some cases) were provided by Khalique and Ntsime [14] through Noether point symmetry approach. Singh et al. [15] applied the modified homotopy analysis method for solving this equation for cases [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Parand et al. gained numerical solutions by using Rational Legendre Pseudospectral approach [16]. The variational iteration method for Lane-Emden singular type equations was employed by Yildirim and Özis [17]. Iqbal and Javed [18] presented a new powerful semianalytic technique by optimal homotopy asymptotic method and obtained numerical solutions for some cases. Numerical solution for this equation was presented by Pandey and Kumar [19] where they applied Bernstein operational matrix of differentiation. Bhrawy and Alofi [20] established numerical solutions for some types of Lane-Emden equation through Jacobi-Gauss collocation method. Heydari et al. [21] employed radial basis functions and used integral operator to solve this equation for some various types. A new shifted second kind chebyshev operational matrix of derivatives was introduced by Doha et al. [22] where they reduced the Lane-Emden type equations to a system of algebraic equations and gained numerical results for a variety of cases. Nazari-Golshan et al. [23] performed a modified homotopy perturbation method coupled with the Fourier transform and solved three different singular nonlinear Lane-Emden equations.

In exploring the analytic solution of Lane-Emden equations by ADM, some calculation problems may occur. In this work we incorporate the spectral method and ADM to overcome these difficulties. By using this method, the time consumption will be reduced, and under some conditions the spectral Adomian decomposition method can be proved to be convergent. In this paper the hybrid spectral Adomian decomposition method will be described in Section 2, while in Section 3, the convergence of ADM and SADM will be presented. The test problem will be interpreted and also the obtained results will be compared by some others method in Section 4, and finally in Section 5, the conclusion in details is provided.

2. Hybrid Spectral Adomian Decomposition Method

2.1. Adomian Decomposition Method for Lane-Emden Equation

The Adomian decomposition method (ADM) is a semianalytical method for ordinary and partial nonlinear differential equations. The details of this method are presented by Adomian [24]. The ADM presented the equation in an operator form by considering the highest-ordered of derivative in the problem. However, a slight change is necessary to overcome the singularity behavior at [figure omitted; refer to PDF] . Hence, in this problem we choose the differential operator [figure omitted; refer to PDF] in terms of the two derivatives, [figure omitted; refer to PDF] ; (1) can be rewritten in the following form: [figure omitted; refer to PDF] where the differential operator [figure omitted; refer to PDF] is [figure omitted; refer to PDF] The inverse operator [figure omitted; refer to PDF] is [figure omitted; refer to PDF] Operating with [figure omitted; refer to PDF] on (3), it follows [figure omitted; refer to PDF] For the solution [figure omitted; refer to PDF] , the ADM introduces an infinite series [figure omitted; refer to PDF] and the infinite series of polynomials [figure omitted; refer to PDF] for the nonlinear term [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , the Adomian polynomials, are obtained as follows: [figure omitted; refer to PDF] By setting (7) and (8) in (6), [figure omitted; refer to PDF] will be obtained. To specify the components [figure omitted; refer to PDF] , ADM which indicates the use of recursive relation will be applied: [figure omitted; refer to PDF] which gives [figure omitted; refer to PDF] The series solution is [figure omitted; refer to PDF] and the [figure omitted; refer to PDF] -term approximation of the series solution will be denoted as [figure omitted; refer to PDF] . This method has been used to solve different equations; see [24, 25].

2.2. Chebyshev Polynomials

Chebyshev polynomials of the first kind are orthogonal with respect to the weight function [figure omitted; refer to PDF] on the interval [figure omitted; refer to PDF] and satisfy the following recursive formula: [figure omitted; refer to PDF] This system is orthogonal basis with weight function [figure omitted; refer to PDF] and orthogonality property: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the Kronecker delta function.

A function [figure omitted; refer to PDF] can be expanded by Chebyshev polynomial as follows: [figure omitted; refer to PDF] where the coefficients [figure omitted; refer to PDF] are [figure omitted; refer to PDF] Here, [figure omitted; refer to PDF] is the inner product of [figure omitted; refer to PDF] . The grid (interpolation) points are chosen to be the extrema [figure omitted; refer to PDF] of the [figure omitted; refer to PDF] th order Chebyshev polynomials [figure omitted; refer to PDF] . The following approximation of the function [figure omitted; refer to PDF] can be introduced: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are the Chebyshev coefficients. These coefficients are determined as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

The Chebyshev polynomials are widely used for numerical solutions of differential, integral, and integrodifferential equations; see [21, 22].

2.3. The Methodology

At first, based on initial condition (2), the initial approximation [figure omitted; refer to PDF] is selected. By applying iteration formula (12), the following will be obtained: [figure omitted; refer to PDF] From (18), the function [figure omitted; refer to PDF] on [figure omitted; refer to PDF] can be approximated as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are the Chebyshev coefficients which are determined from (19) as follows: [figure omitted; refer to PDF] For finding the unknown coefficient [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , by substituting the grid points [figure omitted; refer to PDF] , [figure omitted; refer to PDF] in (21), the following will be concluded: [figure omitted; refer to PDF] From (23) and (24), [figure omitted; refer to PDF] can be gained. Therefore, from (22) and (25) the approximation of [figure omitted; refer to PDF] can be obtained.

For finding the approximation of [figure omitted; refer to PDF] , from (12) the following will be gained: [figure omitted; refer to PDF] In a similar way, the function [figure omitted; refer to PDF] on [figure omitted; refer to PDF] can be approximated as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Similarly, for finding the unknown coefficient [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , by substituting the grid points [figure omitted; refer to PDF] in (21), [figure omitted; refer to PDF] can be concluded; therefore, from (28) and (29) [figure omitted; refer to PDF] will be gained. From (27) and (30) the approximation of [figure omitted; refer to PDF] can be obtained.

Generally, for [figure omitted; refer to PDF] , according to the above method, the approximation of [figure omitted; refer to PDF] will be achieved as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] At the end, [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] term approximation of the series solution.

3. Convergence

3.1. Convergence of ADM

According to [26], (12) can be rewritten as follows: [figure omitted; refer to PDF] If [figure omitted; refer to PDF] then [figure omitted; refer to PDF] Let the function [figure omitted; refer to PDF] in (3) satisfy a Lipschitz condition with Lipschitz constant [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is an upper bound for the above function [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; consequently the following theorem can be achieved.

Theorem 1.

The series solution (7) of problem (1) applying ADM converges if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] ; see [27].

Theorem 2.

The maximum absolute truncation error of series solution (7) to problem (1) is estimated to be [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; see [27].

3.2. Convergence of Spectral Method

Theorem 3.

Let [figure omitted; refer to PDF] (Sobolev space) and [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a positive constant, which depends on a selected norm and is independent of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; see [28].

3.3. Convergence of SADM

Since [figure omitted; refer to PDF] , the series solution of problem (1) using ADM for [figure omitted; refer to PDF] -terms, is [figure omitted; refer to PDF] and also [figure omitted; refer to PDF] , the series solution of problem (1) using SADM for [figure omitted; refer to PDF] -terms, is [figure omitted; refer to PDF] SADM can be shown to be convergent by the following theorem.

Theorem 4.

The series solution (37) of problem (1) using SADM converges if [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Proof.

Consider [figure omitted; refer to PDF] to be the exact solution of (1); then [figure omitted; refer to PDF] According to Theorem 1, [figure omitted; refer to PDF] Also, according to Theorem 3, [figure omitted; refer to PDF] ; Therefore, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . From (39) and (40) the result will be [figure omitted; refer to PDF] Thus, with the increase in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is convergent to [figure omitted; refer to PDF] .

4. Test Problems

Example 1 (see [3, 19]).

Isothermal gas spheres equation: [figure omitted; refer to PDF] with initial conditions [figure omitted; refer to PDF] A series solution obtained by Wazwaz [3] using ADM is [figure omitted; refer to PDF] . This example is solved by the SADM with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The numerical results are shown in Table 1, and it is compared with the ADM [3] and Bernstein operational matrix (BOM) [19].

Table 1: Comparison of the numerical result of Example 1 by present method ( [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ) and numerical values [3, 19].

[figure omitted; refer to PDF]	BOM [19]	RK4	SADM	ADM [3]	Error (SADM)	Error (ADM)
0	0	0	0	0	0	0
0.1	-0.0016658338	-0.0016658338	-0.0016658339	-0.0016658339	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.2	-0.0066533671	-0.0066533659	-0.0066533671	-0.0066533671	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.3	[figure omitted; refer to PDF]	-0.0149328824	-0.0149328833	-0.0149328833	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.4	[figure omitted; refer to PDF]	-0.0264554772	-0.0264554763	-0.0264554763	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.5	-0.0411541005	-0.0411539557	-0.0411539573	-0.0411539568	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.6	[figure omitted; refer to PDF]	-0.0589440694	-0.0589440748	-0.0589440720	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.7	[figure omitted; refer to PDF]	-0.0797260141	-0.0797260042	-0.0797599229	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.8	[figure omitted; refer to PDF]	-0.1033860526	-0.1033860532	-0.1033860110	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.9	[figure omitted; refer to PDF]	-0.1297985460	-0.1297985243	-0.1297983987	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1	-0.1588624338	-0.1588276877	-0.1588276775	-0.1588273537	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Example 2 (see [3, 19]).

Consider the following Lane-Emden equation: [figure omitted; refer to PDF] with initial conditions [figure omitted; refer to PDF]

Case 1 (for [figure omitted; refer to PDF] ).

A series solution obtained by Wazwaz [3] using ADM is [figure omitted; refer to PDF] . This example is solved by the SADM with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The numerical results are shown in Table 2, and it is compared with the BOM [19]. The residual error is illustrated in Figure 1(a), which proves the proposed method is of high accuracy.

Table 2: Comparison of the numerical result of Example 2, (Case 1), by present method ( [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ) and numerical values [19].

[figure omitted; refer to PDF]	BOM [19]	SADM	ADM [3]	Error (BOM)	Error (SADM)	Error (ADM)
0	1	1	1	0	0	0
0.1	0.99833582	0.99833583	0.99833583	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.2	[figure omitted; refer to PDF]	0.99337309	0.99337332	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.3	[figure omitted; refer to PDF]	0.98519979	0.98520236	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.4	[figure omitted; refer to PDF]	0.97395826	0.97397252	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.5	0.95983907	0.95983907	0.95989274	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.6	[figure omitted; refer to PDF]	0.94307317	0.94323076	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.7	[figure omitted; refer to PDF]	0.92392284	0.92431254	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.8	[figure omitted; refer to PDF]	0.90267209	0.90352147	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.9	[figure omitted; refer to PDF]	0.78961717	0.88129745	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1	0.85505757	0.85505757	0.85813584	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Figure 1: (a) Example 2, Case 1: Horizontal axis is related number of iteration ( [figure omitted; refer to PDF] ) and vertical axis shows residual error in mod logarithmic. (b) Example 3: Horizontal axis is related number of iteration ( [figure omitted; refer to PDF] ) and vertical axis shows residual error.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Case 2 (for [figure omitted; refer to PDF] ).

The exact solution is [figure omitted; refer to PDF] . A series solution obtained by Wazwaz [3] using ADM is [figure omitted; refer to PDF] . This example is solved by the SADM with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The numerical results are shown in Table 3, while the absolute error is illustrated in Figure 2 and it is compared with absolute error expressed by BOM [19] which shows that the proposed method is more accurate.

Table 3: The numerical results for Example 2, Case 2, for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	SADM	Exact	ADM [3]	Absolute error (SADM)	Absolute error (ADM)
0	1	1	1	0	0
0.1	0.998337489	0.998337489	0.99833750	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.2	0.993399268	0.993399268	0.99339998	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.3	0.985329278	0.985329278	0.98533726	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.4	0.974354704	0.974354704	0.97439865	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.5	0.960768923	0.960768923	0.96093234	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.6	0.944911183	0.944911183	0.94538460	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.7	0.927145541	0.927145541	0.92829868	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.8	0.907841299	0.907841299	0.91031357	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.9	0.887356510	0.887356509	0.89216242	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1	0.866025411	0.866025404	0.87467085	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Figure 2: Absolute error for Example 2, Case 2. (a) Absolute error obtained by SADM ( [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ). (b) Absolute error obtained by BOM [19].

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Example 3 (see [3]).

Consider the following Lane-Emden equation: [figure omitted; refer to PDF] with initial conditions [figure omitted; refer to PDF] A series solution obtained by Wazwaz [3] using ADM is [figure omitted; refer to PDF] + [figure omitted; refer to PDF] + [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . This example is solved by the SADM with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] on [figure omitted; refer to PDF] . The numerical results are demonstrated in Table 4, and it is compared with the ADM [3]; the residual error is illustrated in Figure 1(b), which proves the proposed method is of high accuracy.

Table 4: Comparison of the numerical result of Example 3 by present method ( [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ) and ADM [3].

[figure omitted; refer to PDF]	RK4	SADM	ADM [3]	Error (SADM)	Error (ADM)
0.1	0.99859793	0.99859793	0.99859793	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.2	0.99439626	0.99439626	0.99439626	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.5	0.96517781	0.96517778	0.96517778	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.8	0.91183204	0.91183203	0.91183204	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1.1	0.83609012	0.83609015	0.83609012	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1.3	0.77443518	0.77443523	0.77443518	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1.5	0.70504518	0.70504523	0.70504518	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1.6	0.66783443	0.66783449	0.66783443	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1.7	0.62917734	0.62917741	0.62917734	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1.8	0.58926208	0.58926216	0.58926208	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1.9	0.54828781	0.54828789	0.54828781	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
2	0.50646355	0.50646362	0.50646355	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Example 4.

Consider the following Lane-Emden equation: [figure omitted; refer to PDF] with initial conditions [figure omitted; refer to PDF]

This example is solved by the SADM with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The numerical results are shown in Table 5, and the absolute errors are illustrated in Figure 3. The exact solution is [figure omitted; refer to PDF] ; see [14].

Table 5: The numerical results for Example 4 for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	SADM	Exact	Absolute error
0.1	-0.01990066	-0.01990066	[figure omitted; refer to PDF]
0.2	-0.07844143	-0.07844143	[figure omitted; refer to PDF]
0.3	-0.17235539	-0.17235539	[figure omitted; refer to PDF]
0.4	-0.29684001	-0.29684001	[figure omitted; refer to PDF]
0.5	-0.44628707	-0.44628710	[figure omitted; refer to PDF]
0.6	-0.61496760	-0.61496940	0.00000180
0.7	-0.79750315	-0.79755224	0.00004904
0.8	-0.98854610	-0.98939248	0.00084639

Figure 3: (a) Example 4: Horizontal axis is related number of iteration ( [figure omitted; refer to PDF] ) and vertical axis shows absolute error. (b) Example 5: Horizontal axis is related number of iteration ( [figure omitted; refer to PDF] ) and vertical axis shows absolute error in mod logarithmic.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Example 5 (see [3]).

Consider the following Lane-Emden equation: [figure omitted; refer to PDF] with initial conditions [figure omitted; refer to PDF] A series solution obtained by Wazwaz [3] using ADM is [figure omitted; refer to PDF] . This example is solved by the SADM with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The numerical results are shown in Table 6, and the absolute error is illustrated in Figure 3. The exact solution is [figure omitted; refer to PDF] ; see [14].

Table 6: Comparison of the numerical result of Example 5 by present method ( [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ), ADM [3], and exact solution.

[figure omitted; refer to PDF]	SADM	ADM [3]	Exact	Absolute error (SADM)	Absolute error (ADM)
0.1	1.01005017	1.01005017	1.01005017	[figure omitted; refer to PDF]	0
0.2	1.04081077	1.04081078	1.04081077	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.3	1.09417428	1.09417428	1.09417428	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.4	1.17351087	1.17351085	1.17351087	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.5	1.28402542	1.28402507	1.28402542	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.6	1.43332941	1.43332623	1.43332941	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.7	1.63231622	1.63229556	1.63231622	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.8	1.89648088	1.89637596	1.89648088	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Example 6 (see [3]).

Consider the following Lane-Emden equation: [figure omitted; refer to PDF] with initial conditions [figure omitted; refer to PDF] A series solution obtained by Wazwaz [3] using ADM is [figure omitted; refer to PDF] + [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . This example is solved by the SADM with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , 0.4, 0.6, and 0.8, for which the results are shown in Tables 7-10, respectively, and they are compared with ADM [3].

Table 7: The numerical results for Example 6 for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	ADM [3]	SADM	RK4	Residual error
0.1	0.99880903	0.99880903	0.99880903	[figure omitted; refer to PDF]
0.2	0.99525519	0.99525519	0.99525519	[figure omitted; refer to PDF]
0.3	0.98939500	0.98939500	0.98939500	[figure omitted; refer to PDF]
0.4	0.98132029	0.98132028	0.98132028	[figure omitted; refer to PDF]
0.5	0.97115486	0.97115480	0.97115479	[figure omitted; refer to PDF]
0.6	0.95905026	0.95904988	0.95904987	[figure omitted; refer to PDF]
0.7	0.94518104	0.94517924	0.94517923	[figure omitted; refer to PDF]
0.8	0.92574024	0.92973345	0.92973343	[figure omitted; refer to PDF]
0.9	0.91293603	0.91291419	0.91291416	[figure omitted; refer to PDF]
1	0.89499072	0.89492872	0.89492869	[figure omitted; refer to PDF]

Table 8: The numerical results for Example 6 for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	ADM [3]	SADM	RK4	Residual error
0.1	0.99922630	0.99922630	0.99922630	[figure omitted; refer to PDF]
0.2	0.99691595	0.99691595	0.99691595	[figure omitted; refer to PDF]
0.3	0.99310085	0.99310085	0.99310085	[figure omitted; refer to PDF]
0.4	0.98783306	0.98783306	0.98783306	[figure omitted; refer to PDF]
0.5	0.98118327	0.98118325	0.98118324	[figure omitted; refer to PDF]
0.6	0.97323871	0.97323859	0.97323858	[figure omitted; refer to PDF]
0.7	0.96410081	0.96410027	0.96410026	[figure omitted; refer to PDF]
0.8	0.95388278	0.95388075	0.95388074	[figure omitted; refer to PDF]
0.9	0.94270739	0.94270085	0.94270083	[figure omitted; refer to PDF]
1	0.93070541	0.93068681	0.93068679	[figure omitted; refer to PDF]

Table 9: The numerical results for Example 6 for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	ADM [3]	SADM	RK4	Residual error
0.1	0.99957876	0.99957876	0.99957876	[figure omitted; refer to PDF]
0.2	0.99831983	0.99831983	0.99831983	[figure omitted; refer to PDF]
0.3	0.99623743	0.99623743	0.99623743	[figure omitted; refer to PDF]
0.4	0.99335491	0.99335491	0.99335491	[figure omitted; refer to PDF]
0.5	0.98970415	0.98970415	0.98970415	[figure omitted; refer to PDF]
0.6	0.98532490	0.98532488	0.98532487	[figure omitted; refer to PDF]
0.7	0.98026385	0.98026375	0.98026374	[figure omitted; refer to PDF]
0.8	0.97457371	0.97457333	0.97457332	[figure omitted; refer to PDF]
0.9	0.96831224	0.96831099	0.96831098	[figure omitted; refer to PDF]
1	0.96154132	0.96153777	0.96153776	[figure omitted; refer to PDF]

Table 10: The numerical results for Example 6 for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	ADM [3]	SADM	RK4	Residual error
0.1	0.99985103	0.99985103	0.99985103	[figure omitted; refer to PDF]
0.2	0.99940531	0.99940531	0.99940531	[figure omitted; refer to PDF]
0.3	0.99866642	0.99866642	0.99866642	[figure omitted; refer to PDF]
0.4	0.99764024	0.99764024	0.99764025	[figure omitted; refer to PDF]
0.5	0.99633490	0.99633490	0.99633489	[figure omitted; refer to PDF]
0.6	0.99476060	0.99476060	0.99476059	[figure omitted; refer to PDF]
0.7	0.99292949	0.99292949	0.99292948	[figure omitted; refer to PDF]
0.8	0.99085553	0.99085550	0.99085549	[figure omitted; refer to PDF]
0.9	0.98855418	0.98855410	0.98855408	[figure omitted; refer to PDF]
1	0.98604231	0.98604208	0.98604207	[figure omitted; refer to PDF]

5. Conclusion

In this paper a new modification of the Adomian decomposition method was presented for solving Lane-Emden equations. This method, named spectral Adomian decomposition method (SADM), was successfully applied to solve the Lane-Emden equations.

It is clear that in computation of integral in the right hand of (12), two difficulties may arise:

(i) In calculation, the Adomian polynomials [figure omitted; refer to PDF] may be so much problematic that the integration becomes very complicated.

(ii) By increasing [figure omitted; refer to PDF] , the number of terms of approximate solution may increase so much rapidly that the integration becomes both complicated and time-consuming.

To overcome the abovementioned difficulties, SADM is proposed, a combination of Adomian decomposition method and Chebyshev pseudo-spectral method. By analyzing and comparing the procedures used in SADM and ADM, it is observed that the new approach overcomes the difficulties (i) and (ii), leading to complicated calculations and time-consuming integrals and terms not necessary in the standard ADM. Moreover, unlike spectral methods, the proposed method does not require the solution of any linear or nonlinear system of equations.

A comparative study between the proposed method and both 4th order Runge-Kutta method (RK4) and ADM was presented in tables. Consequently, the obtained results showed that SADM can solve the problem effectively, while comparison demonstrates that the proposed technique is in good agreement with the numerical results obtained using RK4.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] O. W. Richardson The Emission of Electricity from Hot Bodies , Longmans Green and Co., London, UK, 1921.

[2] A.-M. Wazwaz, "The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models," Communications in Nonlinear Science and Numerical Simulation , vol. 16, no. 10, pp. 3881-3886, 2011.

[3] A.-M. Wazwaz, "A new algorithm for solving differential equations of Lane-Emden type," Applied Mathematics and Computation , vol. 118, no. 2-3, pp. 287-310, 2001.

[4] A.-M. Wazwaz, "A new method for solving singular initial value problems in the second-order ordinary differential equations," Applied Mathematics and Computation , vol. 128, no. 1, pp. 45-57, 2002.

[5] M. I. Nouh, "Accelerated power series solution of polytropic and isothermal gas spheres," New Astronomy , vol. 9, no. 6, pp. 467-473, 2004.

[6] H. Goenner, P. Havas, "Exact solutions of the generalized Lane-Emden equation," Journal of Mathematical Physics , vol. 41, no. 10, pp. 7029-7042, 2000.

[7] S. Liao, "A new analytic algorithm of Lane-Emden type equations," Applied Mathematics and Computation , vol. 142, no. 1, pp. 1-16, 2003.

[8] J.-H. He, "Variational approach to the Lane-Emden equation," Applied Mathematics and Computation , vol. 143, no. 2-3, pp. 539-541, 2003.

[9] S. A. Yousefi, "Legendre wavelets method for solving differential equations of Lane-Emden type," Applied Mathematics and Computation , vol. 181, no. 2, pp. 1417-1422, 2006.

[10] E. Momoniat, C. Harley, "Approximate implicit solution of a Lane-Emden equation," New Astronomy , vol. 11, no. 7, pp. 520-526, 2006.

[11] A. Yildirim, T. Özis, "Solutions of singular IVPs of Lane-Emden type by homotopy perturbation method," Physics Letters, Section A: General, Atomic and Solid State Physics , vol. 369, no. 1-2, pp. 70-76, 2007.

[12] M. S. H. Chowdhury, I. Hashim, "Solutions of a class of singular second-order IVPs by homotopy-perturbation method," Physics Letters A , vol. 365, no. 5-6, pp. 439-447, 2007.

[13] J. I. Ramos, "Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method," Chaos, Solitons & Fractals , vol. 38, no. 2, pp. 400-408, 2008.

[14] C. M. Khalique, P. Ntsime, "Exact solutions of the Lane-Emden-type equation," New Astronomy , vol. 13, no. 7, pp. 476-480, 2008.

[15] O. P. Singh, R. K. Pandey, V. K. Singh, "An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified homotopy analysis method," Computer Physics Communications , vol. 180, no. 7, pp. 1116-1124, 2009.

[16] K. Parand, M. Shahini, M. Dehghan, "Rational Legendre pseudospectral approach for solving nonlinear differential equations of LANe-Emden type," Journal of Computational Physics , vol. 228, no. 23, pp. 8830-8840, 2009.

[17] A. Yildirim, T. Özis, "Solutions of singular IVPs of Lane-Emden type by the variational iteration method," Nonlinear Analysis: Theory, Methods & Applications , vol. 70, no. 6, pp. 2480-2484, 2009.

[18] S. Iqbal, A. Javed, "Application of optimal homotopy asymptotic method for the analytic solution of singular Lane-Emden type equation," Applied Mathematics and Computation , vol. 217, no. 19, pp. 7753-7761, 2011.

[19] R. K. Pandey, N. Kumar, "Solution of Lane-Emden type equations using Bernstein operational matrix of differentiation," New Astronomy , vol. 17, no. 3, pp. 303-308, 2012.

[20] A. H. Bhrawy, A. S. Alofi, "A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations," Communications in Nonlinear Science and Numerical Simulation , vol. 17, no. 1, pp. 62-70, 2012.

[21] M. Heydari, S. M. Hosseini, G. B. Loghmani, "Numerical solutions of singular IVPs of lane emden type using integral operator and radial basis functions," International Journal of Industrial Mathematics , vol. 4, no. 2, 2012.

[22] E. H. Doha, W. M. Abd- Elhameed, Y. H. Youssri, "Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type," New Astronomy , vol. 23-24, pp. 113-117, 2013.

[23] A. Nazari-Golshan, S. S. Nourazar, H. Ghafoori-Fard, A. Yildirim, A. Campo, "A modified homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane-Emden equations," Applied Mathematics Letters , vol. 26, no. 10, pp. 1018-1025, 2013.

[24] G. Adomian Nonlinear Stochastic Operator Equations , Academic Press, New York, NY, USA, 1986.

[25] S. Abbasbandy, "A numerical solution of Blasius equation by Adomian's decomposition method and comparison with homotopy perturbation method," Chaos, Solitons & Fractals , vol. 31, no. 1, pp. 257-260, 2007.

[26] A.-M. Wazwaz A First Course in Integral Equations , World Scientific, 1997.

[27] I. L. El-Kalla, "Convergence of the Adomian method applied to a class of nonlinear integral equations," Applied Mathematics Letters , vol. 21, no. 4, pp. 372-376, 2008.

[28] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang Spectral Methods in Fluid Dynamics , Springer, New York, NY, USA, 1988.