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.
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.
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.
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.
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.
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).