ISRN Mathematical PhysicsVolume 2013 (2013), Article ID 507145, 9 pageshttp://dx.doi.org/10.1155/2013/507145

Research Article

Numerical Solution of Singular Lane-Emden Equation

Hossein Aminikhah and Sakineh Moradian

Department of Applied Mathematics, School of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht 41938, Iran

Received 24 June 2013; Accepted 13 July 2013

Academic Editors: N. E. Grandi and R. Schiappa

Copyright © 2013 Hossein Aminikhah and Sakineh Moradian. 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.

Abstract.

A new approach for solving the nonlinear Lane-Emden type equations has been proposed. The method is based on Legendre wavelets approximations. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.

1. Introduction

The Lane-Emden type equations are nonlinear ordinary differential equations on semi-infinite domain. They are categorized as singular initial value problems. These equations describe the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of classical thermodynamics. The polytrophic theory of stars essentially follows out of thermodynamic considerations that deal with the issue of energy transport, through the transfer of material between different levels of the star. These equations are one of the basic equations in the theory of stellar structure and have been the focus of many studies. The general form of the Lane-Emden equations is the following form: with the following initial conditions: where is a continuous real-value function and   is an analytical function. Equation (1) was used to model several phenomena in mathematical physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas sphere, and theory of thermionic currents [1, 2]. The solution of the Lane-Emden equation, as well as those of a variety of nonlinear problems in quantum mechanics and astrophysics such as the scattering length calculations in the variable phase approach, is numerically challenging because of the singular point at the origin. Bender et al. [3] proposed a new perturbation technique based on an artificial parameter  ; the method is often called -method. El-Gebeily and O’Regan [4] used the quasilinearization approach to solve the standard Lane-Emden equation. This method approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories, it is not based on the existence of some small parameters. Approximate solutions to the above problems were presented by Shawagfeh [5] and Wazwaz [6, 7] by applying the Adomian method which provides a convergent series solution. Nouh [8] accelerated the convergence of a power series solution of the Lane-Emden equation by using an Euler-Abel transformation and Padé approximation. Mandelzweig and Tabakin [9] applied Bellman and Kalaba’s quasilinearization method, and Ramos [10] used a piecewise linearization technique based on the piecewise linearization of the Lane-Emden equation. Bozkhov and Gilli Martins [11] and later Momoniat and Harley [12] applied the Lie Group method successfully to generalized Lane-Emden equations of the first kind. Exact solutions of generalized Lane-Emden solutions of the first kind are investigated by Goenner and Havas [13]. Liao [14] solved Lane-Emden type equations by applying a homotopy analysis method. He [15] obtained an approximate analytical solution of the Lane-Emden equation by applying a variational approach which uses a semi-inverse method. Ramos [16] presented a series approach to the Lane-Emden equation and gave the comparison with homotopy perturbation method. Özis and Yildirim [17, 18] gave the solutions of a class of singular second-order IVPs of Lane-Emden type by using homotopy perturbation and variational iteration method. Parand et al. [19–22] presented three numerical techniques to solve higher ordinary differential equations such as Lane-Emden. Their approach was based on the rational Chebyshev, rational Legendre Tau, and Hermite functions collocation methods. In this paper, the new approximate analytical method will be introduced for exact solution of Lane-Emden equation.

This paper is arranged as follows.

In Section 2, the properties of Legendre wavelets and the way to construct the wavelet technique for this type of equation are described. In Section 3, the proposed method is applied to some types of Lane-Emden equations, and a comparison is made with the existing analytic or exact solutions that were reported in other published works in the literature. Finally, we give a brief conclusion in the last section.

2. Legendre Wavelets Applied to Singular IVPs of Lane-Emden Type Equation

2.1. Wavelets and Legendre Wavelets

Wavelets constitute a family of functions constructed from dilation and translation of single function called the mother wavelet. When the dilation parameter and the translation parameter vary continuously, we have the following family of continuous wavelets:

If we restrict the parameter and   to discrete values as  , and    positive integers, we have the following family of discrete wavelets: where forms a wavelet basis for . In particular, when   and    then forms an orthonormal basis.

Legendre wavelets    have four arguments,   can assume any positive integer, is the order for Legendre polynomials, and they are defined on the interval as follows: where   , . The coefficient is for orthonormality. Here, are the well-known Legendre polynomials of order which are defined on the interval and can be determined with the aid of the following recurrence formulae:

2.2. Function Approximation

A function defined on the interval may be expanded by Legendre wavelet as where

In (8), denotes the inner product.

If the infinite series in (7) is truncated, then (7) can be written as where and are matrices given by

The integration of the product of two Legendre wavelets vector functions is obtained as where    is an identity matrix.

2.3. Legendre Wavelets Operational Matrix of Integration

The integration of the vectordefined in (11) can be obtained as where    is the operational matrix of integration given by [23] as where, and are matrices given by

The following property of the product of two Legendre wavelet vector functions will also be used: where is a vector given in (10) and is a matrix, which is called the product operation of Legendre wavelet vector functions [23]. For   and  , the matrix is obtained:

2.4. Solution of Lane-Emden Equations

We multiply both sides of (1) by , in order to use Legendre wavelets to approximate as

Integrating (19) with respect to twice from 0 to  , we obtain where coefficients and are known and can be obtained from the initial conditions, and are defined similarly to (10) and (11), and is operational matrix for integration, defined in (14).

Also consider the following approximations: where   and   are column vectors with the entries of the vectors   and coefficients of   known.

Substitution of approximations (20) and (22) into (18) will be resulted to

Simplifying     in (23), a nonlinear system in terms of    will be obtained:

The element of vector functions    can be computed by solving these systems.

3. Numerical Examples

In this section, some examples of Lane-Emden equation are considered and will be solved by introduced method.

Example 1. Consider the following nonlinear Lane-Emden equation: subject to the initial conditions which has the following analytical solution: We solve (25) by the method discussed in this paper with    and  .We multiply both sides of (25) by  ,

Let us consider the following approximations:

Substitution into (28) and simplifying will be resulted to:

By solving the system (30), we have

Therefore, the approximate solution of (25) will be obtained as follows:

Table 1 shows some values of the solutions and absolute errors at some , and plots of the exact and approximate solutions are shown in Figure 1.

Table 1: Numerical results of Example 1.

Exact solution
Legendre wavelets
Absolute error

0.0
1
1.000020858
0.000020858

0.1
0.9900498337
0.9900449375
0.0000048962

0.2
0.9607894392
0.9607962885
0.0000068493

0.3
0.9139311853
0.9139303816

0.4
0.8521437890
0.8521354021
0.0000083869

0.5
0.7788007831
0.7787879122
0.0000128709

0.6
0.6976763261
0.6976230812
0.0000532449

0.7
0.6126263942
0.6124194854
0.0002069088

0.8
0.5272924240
0.5266984761
0.0005939479

0.9
0.4448580662
0.4434381168
0.0014199494

1
0.3678794412
0.3648016892
0.0030777520

Figure 1: The exact and LWM solution of Example 1.

Example 2. Consider the following nonlinear Lane-Emden equation: subject to the initial conditions where   is constant. Substituting  , and    into (33) leads to the exact solution respectively.

For  , we solve (33) by the Legendre wavelet method with and . For this equation, we find

Therefore, we have which is the exact solution.

For  , we solve (33) by the method discussed in this paper with   and  . We have

Therefore, the following solution will result:

Table 2 shows that the Legendre wavelet solution is very near to the exact solution. Figure 2(a) shows that Legendre wavelet solution coincides with the exact solution.

Table 2: Numerical results of Example 2 for .

Exact solution
Legendre wavelets
Absolute error

0.0
1
1
0

0.1
0.9983341665
0.9983341665
0

0.2
0.9933466540
0.9933466540
0

0.3
0.9850673556
0.9850673556
0

0.4
0.9735458558
0.9735458558
0

0.5
0.9588510772
0.9588510772
0

0.6
0.9410707892
0.9410707890

0.7
0.0920310982
0.0920310989

0.8
0.8966951136
0.8966951137

0.9
0.8703632328
0.8703632329

1
0.8414709848
0.8414709848
0

(a)

(b)

Figure 2: (a) The exact and LWM solution of Example 2 for  . (b) The exact and LWM solution of Example 2 for  .

For solving (33) by Legendre wavelets with  , and  ,  we find

The approximate solution of is as follows:

Table 3 shows that the Legendre wavelet solution is very near to the exact solution. Figure 2(b) shows that Legendre wavelet solution coincides with the exact solution.

Table 3: Numerical results of Example 2 for .

Exact solution
Legendre wavelets
Absolute error

0.0
1
1
0

0.1
0.9983374884
0.9983374884
0

0.2
0.9933992679
0.9933992677

0.3
0.9853292781
0.9853292781
0

0.4
0.9743547036
0.9743547036
0

0.5
0.9607689228
0.9607689228
0

0.6
0.9449111826
0.9449111825

0.7
0.9271455411
0.9271455408

0.8
0.9078412992
0.9078412989

0.9
0.8873565093
0.8873565094

1
0.8660254038
0.8660254038
0

Example 3. Consider the following nonlinear Lane-Emden equation: subject to the initial conditions which has the following analytical solution: Solving (42) by Legendre wavelets method with    and  , we have Therefore, the solution of the Lane-Emden equation will be obtained as follows:

Table 4 shows that the Legendre wavelet solution is very near to the exact solution. Figure 3 shows that Legendre wavelet solution coincides with the exact solution.

Table 4: Numerical results of Example 3.

Exact solution
Legendre wavelets
Absolute error

0.0
1
0.9999999958

0.1
1.010050167
1.010050166

0.2
1.040810774
1.040810774
0

0.3
1.094174284
1.094174282

0.4
1.173510871
1.173510871
0

0.5
1.284025417
1.284025415

0.6
1.433329415
1.433329414

0.7
1.632316220
1.632316219

0.8
1.896480879
1.896480878

0.9
2.247907987
2.247907986

1
2.718281828
2.718281824

Figure 3: The exact and LWM solution of Example 3.

Example 4. Consider the following nonlinear Lane-Emden equation: subject to the initial conditions which has the following analytical solution:

We solve (47) by the method discussed in this paper with    and  . We multiply both sides of (47) by  ,

Let us consider the following approximations:

Substitution into (50) and simplifying will be resulted to

By solving the system (52), we have

The approximate solution of is as follows:

Table 5 shows some values of the solutions and absolute errors at some , and plots of the exact and approximate solutions are shown in Figure 4.

Table 5: Numerical results of Example 4.

Exact solution
Legendre wavelets
Absolute error

0.0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4: The exact and LWM solution of Example 4.

Example 5. Consider the following nonlinear Lane-Emden equation: subject to the initial conditions which has the following analytical solution:

We solve (55) by the method discussed in this paper with and . This implies that

The approximate solution of is as follows:

Table 6 shows that the Legendre wavelet solution is very near to the exact solution. Figure 5 shows that Legendre wavelet solution coincides with the exact solution.

Table 6: Numerical results of Example 5.

Exact solution
Legendre wavelets
Absolute error

0.0
0

0.1

0.2

0.3

0

0.4

0.5

0.6

0.7

0.8

0.9

1
0

Figure 5: The exact and LWM solution of Example 5.

4. Conclusion

In this research, we have presented the Legendre wavelet method for solving nonlinear singular Lane-Emden equation. The Legendre wavelets operational matrix of integration is used to solve Lane-Emden equation. The present method reduces Lane-Emden equation into a set of algebraic equations. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.

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