Abstract and Applied Analysis

Volume 2014 (2014), Article ID 735831, 8 pages

http://dx.doi.org/10.1155/2014/735831

Research Article

Using an Effective Numerical Method for Solving a Class of Lane-Emden Equations

Yulan Wang,¹ Hao Yu,¹ Fugui Tan,² and Shuguang Li¹

¹Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, China

²Jining Teachers College, Wūlánchábù, Inner Mongolia 012000, China

Received 26 March 2014; Revised 12 May 2014; Accepted 22 May 2014; Published 18 June 2014

Academic Editor: Igor Leite Freire

Copyright © 2014 Yulan Wang et al. 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.

We use the reproducing kernel method to solve the well-known classes of Lane-Emden-type equations. These classes of equations have the form of Lane-Emden problem. Comparing the results of the reproducing kernel method with the analytical solutions by means of some typical examples, we can affirm that the reproducing kernel method is an efficient and accurate method.

1. Introduction

Let us consider the following Lane-Emden problem:

where , is given bounded, continuous function, and is nonlinear function; in [1], we can see that the most popular form of is , where is a constant parameter; this type of equation is the Lane-Emden equations of the first kind; in addition, can be the exponential functions ; this type of equations is called the Lane-Emden equations of the second kind; furthermore, the function can be logarithmic functions and trigonometric functions; all these types of equations are named after the astrophysicists Jonathan Lane and Robert Emden; they were the first to study these types of equations. Lane-Emden equations are widely used in various physical phenomena. Many scholars [2–5] devote their energies to this field, with the high development of computer technology; lots of numerical methods have been put forward to solve this type of equation, such as pseudospectral method, Haar wavelet method, and Adomian decomposition method (ADM) [6–11].

Reproducing kernel method (RKM) is an attractive method because of its accuracy, and it has already been applied to various fields. In this paper, we use the reproducing kernel method to solve (1) to show the efficiency and accuracy of this method.

2. The Reproducing Kernel Method

2.1. Practise Homogenization for Lane-Emden Equations

In order to use reproducing kernel method to solve (1), we need to practise homogenization for (1); previously, we multiplied (1) by ; we find that

where . Obviously, the solution of (2) is the solution of (1). So we only need to gain the solution of (2). The question (2) with nonhomogeneous boundary value conditions is equivalent to the problem of having a function satisfying

where .

2.2. Construct Reproducing Kernel Space

Aiming at the purpose of solving (3), we need to introduce the reproducing kernel space; previously, let us introduce the concept of the reproducing kernel space.

For each of , there is a function of two variables , where is Hilbert space and is a set abstraction. If we can get

we say that is the reproducing kernel Hilbert space and is the reproducing kernel of .

We give a linear space as follows:

According to [12, 13], we give the inner product as follows:

And according to [14], we can prove that is a reproducing kernel space; its reproducing kernel is

In order to use reproducing kernel method to solve (3) and referring to [15, 16], we can get as follows:

where .

Then practise Gram-Schmidt orthonormalization for ; according to [17, 18] we get

where are coefficients of Gram-Schmidt orthonormalization.

If are distinct points dense in and is existent, we get that

is the solution of (3). The proof of it refers to [19, 20]. If the equations are linear ones, , we can solve the problems directly. If they are nonlinear equations, we have to use iteration method to solve them, and the specific methodology refers to [21, 22].

2.3. The Approximate Solution

We denote the approximate solution of by

According to the proof of [23] we can easily get that and .

3. Numerical Experiment

Example 1. Let us talk about the well-known polytropic differential equation in [3]. Consider

whose exact solution is given by ; using the reproducing kernel method, , , , and . The numerical results are shown in Figure 1 and Table 1.

Table 1: Numerical solutions for Example 1.

0.0
1
1
0
0
0.00298864

0.1
0.998337
0.998338

0.2
0.993399
0.9934

0.3
0.985329
0.98533

0.4
0.974355
0.974355

0.5
0.960769
0.96077

0.6
0.944911
0.944912

0.7
0.927146
0.927147

0.8
0.907841
0.907843

0.9
0.887357
0.887359

1.0
0.866025
0.866029

Figure 1: Present numerical method for Example 1.

Example 2. Considering the following nonlinear equation:

the exact solution is given by ; using the reproducing kernel method, , , , and . The numerical results are shown in Figure 2 and Table 2.

Table 2: Numerical solutions for Example 2.

0.0
1
1
0
0
0.0112026

0.1
0.995037
0.99504

0.2
0.980581
0.980585

0.3
0.957826
0.957832

0.4
0.928477
0.928485

0.5
0.894427
0.894438

0.6
0.857493
0.857506

0.7
0.819232
0.819258

0.8
0.780869
0.780948

0.9
0.743294
0.74356

1.0
0.707107
0.707909

Figure 2: Present numerical method for Example 2.

Example 3. Consider a linear Lane-Emden equation:

where ; using the reproducing kernel method, , , , and . The exact solution is given by . The numerical results are presented in Figure 3 and Table 3.

Table 3: Numerical solutions for Example 3.

Exact solution
50-order RKM approximation
100-order RKM approximation
150-order RKM approximation

0.0
0
0
0
0

0.1
0.009
0.00886466
0.00897728
0.00899228

0.2
0.032
0.0318207
0.0319718
0.0319906

0.3
0.063
0.0627957
0.0629686
0.0629896

0.4
0.096
0.0957782
0.0959661
0.0959888

0.5
0.125
0.124764
0.124964
0.124988

0.6
0.144
0.143752
0.143962
0.143987

0.7
0.147
0.146741
0.14696
0.146986

0.8
0.128
0.12773
0.127957
0.127985

0.9
0.081
0.0807176
0.0809539
0.0809837

1.0
0
−0.000295352
−0.0000496372
−0.0000179019

Figure 3: Present numerical method for Example 3.

Example 4. Consider a Lane-Emden equation of the second kind in [1]. One has

where ; using the reproducing kernel method, , , , and . The exact solution is given by . The numerical results are presented in Figure 4 and Table 4.

Table 4: Numerical solutions for Example 4.

0.0
0
0
0
0
0.0198013

0.1
0.00499584
0.00497722

0.2
0.0199337
0.0198951

0.3
0.0446665
0.0446089

0.4
0.0789556
0.0788834

0.5
0.122479
0.122407

0.6
0.174846
0.174807

0.7
0.235604
0.235676

0.8
0.304261
0.304599

0.9
0.380294
0.381194

1.0
0.463163
0.465161

Figure 4: Present numerical method for Example 4.

Example 5. Consider a Lane-Emden equation in [3]. One has

Using the reproducing kernel method, , , , and . The exact solution is given by . The numerical results are presented in Figure 5 and Table 5.

Table 5: Numerical solutions for Example 5.

Exact solution
50-order RKM approximation
The approximation solutions in [3]

in [3]

0.0
0
0
0

0.01
−0.0001999900
−0.00019999
−0.0001970587

0.1
−0.0199006617
−0.0199007
−0.0198967225

0.5
−0.4462871026
−0.446287
−0.4462840851

1.0
−1.3862943611
−1.38629
−1.3862934297

2.0
−3.2188758249
−3.21888
−3.2188763248

3.0
−4.6051701860
−4.60517
−4.6051709964

4.0
−5.6664266881
−5.66643
−5.6664274573

Figure 5: Present numerical method for Example 5.

4. Conclusions and Remarks

In this paper, reproducing kernel method has been used to solve some typical Lane-Emden examples; the computation implies that the solutions by the reproducing kernel method are very accurate. Moreover, the first and second derivatives of the solutions also have very high accuracy. From all of this, we can affirm that the reproducing kernel method is an efficient and accurate method. All computations are performed by the Mathematica 8.0 software package.

Conflict of Interests

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

Acknowledgments

The authors thank the reviewers for their valuable suggestions, which greatly improved the quality of the paper. This paper is supported by the Natural Science Foundation of China (no. 11361037), the Natural Science Foundation of Inner Mongolia (no. 2013MS0109), and Project Application Technology Research and Development Foundation of Inner Mongolia (no. 20120312).

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