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.

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 ofsubject to initial conditions:where and are constants, is a continuous real valued function, and . 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 . 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 Öziş [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 and . 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 Yıldırım and Öziş [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 . Hence, in this problem we choose the differential operator in terms of the two derivatives, ; (1) can be rewritten in the following form:where the differential operator isThe inverse operator isOperating with on (3), it followsFor the solution , the ADM introduces an infinite seriesand the infinite series of polynomialsfor the nonlinear term , where , the Adomian polynomials, are obtained as follows:By setting (7) and (8) in (6),will be obtained. To specify the components , ADM which indicates the use of recursive relation will be applied:which givesThe series solution is and the -term approximation of the series solution will be denoted as . 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 on the interval and satisfy the following recursive formula:This system is orthogonal basis with weight function and orthogonality property:where , for and is the Kronecker delta function.

A function can be expanded by Chebyshev polynomial as follows:where the coefficients areHere, is the inner product of . The grid (interpolation) points are chosen to be the extremaof the th order Chebyshev polynomials . The following approximation of the function can be introduced:where are the Chebyshev coefficients. These coefficients are determined as follows:where

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 is selected. By applying iteration formula (12), the following will be obtained:From (18), the function on can be approximated as follows:where are the Chebyshev coefficients which are determined from (19) as follows:For finding the unknown coefficient , , by substituting the grid points , in (21), the following will be concluded:From (23) and (24),can be gained. Therefore, from (22) and (25) the approximation of can be obtained.

For finding the approximation of , from (12) the following will be gained:In a similar way, the function on can be approximated aswhereSimilarly, for finding the unknown coefficient , , by substituting the grid points in (21),can be concluded; therefore, from (28) and (29)will be gained. From (27) and (30) the approximation of can be obtained.

Generally, for , according to the above method, the approximation of will be achieved as follows:whereAt the end, is the term approximation of the series solution.

3. Convergence

3.1. Convergence of ADM

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

Theorem 1. The series solution (7) of problem (1) applying ADM converges if and , where ; see [27].

Theorem 2. The maximum absolute truncation error of series solution (7) to problem (1) is estimated to be where and ; see [27].

3.2. Convergence of Spectral Method

Theorem 3. Let (Sobolev space) and ; then where is a positive constant, which depends on a selected norm and is independent of and ; see [28].

3.3. Convergence of SADM

Since , the series solution of problem (1) using ADM for -terms, isand also , the series solution of problem (1) using SADM for -terms, isSADM can be shown to be convergent by the following theorem.

Theorem 4. The series solution (37) of problem (1) using SADM converges if , , and .

Proof. Consider to be the exact solution of (1); thenAccording to Theorem 1,Also, according to Theorem 3, ; Therefore,where . From (39) and (40) the result will be Thus, with the increase in and , is convergent to .

4. Test Problems

Example 1 (see [3, 19]). Isothermal gas spheres equation:with initial conditionsA series solution obtained by Wazwaz [3] using ADM is . This example is solved by the SADM with and . The numerical results are shown in Table 1, and it is compared with the ADM [3] and Bernstein operational matrix (BOM) [19].

Example 2 (see [3, 19]). Consider the following Lane-Emden equation:with initial conditions

Case 1 (for ). A series solution obtained by Wazwaz [3] using ADM is . This example is solved by the SADM with and . 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.

(a)

(b)

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Figure 1 

(a) Example 2, Case 1: Horizontal axis is related number of iteration () and vertical axis shows residual error in mod logarithmic. (b) Example 3: Horizontal axis is related number of iteration () and vertical axis shows residual error.

Case 2 (for ). The exact solution is . A series solution obtained by Wazwaz [3] using ADM is . This example is solved by the SADM with and . 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.

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Figure 2 

Absolute error for Example 2, Case 2. (a) Absolute error obtained by SADM (, ). (b) Absolute error obtained by BOM [19].

Example 3 (see [3]). Consider the following Lane-Emden equation:with initial conditionsA series solution obtained by Wazwaz [3] using ADM is + + , where and . This example is solved by the SADM with and on . 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.

Example 4. Consider the following Lane-Emden equation:with initial conditions

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

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Figure 3 

(a) Example 4: Horizontal axis is related number of iteration () and vertical axis shows absolute error. (b) Example 5: Horizontal axis is related number of iteration () and vertical axis shows absolute error in mod logarithmic.

Example 5 (see [3]). Consider the following Lane-Emden equation:with initial conditionsA series solution obtained by Wazwaz [3] using ADM is . This example is solved by the SADM with and . The numerical results are shown in Table 6, and the absolute error is illustrated in Figure 3. The exact solution is ; see [14].

Example 6 (see [3]). Consider the following Lane-Emden equation:with initial conditionsA series solution obtained by Wazwaz [3] using ADM is + , where . This example is solved by the SADM with and for , 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].