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

and Applied Analysis 3 0.2 0.4 0.6 0.8 1.0 x 0.75 0.80 0.85 0.90 0.95 1.00


Introduction
Let us consider the following Lane-Emden problem: () +     () +  () =  () ,  ∈ [0, ] , where  ≥ 0, () 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][3][4][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][7][8][9][10][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.
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 V() satisfying where (, V) = () − (V +   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 .
If {  } ∞ =1 are distinct points dense in [0, ] and L −1 is existent, we get that is the solution of (3).The proof of it refers to [19,20].If the equations are linear ones, (, V) = (), 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].

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.

Table 1 :
Numerical solutions for Example 1.

Table 2 :
Numerical solutions for Example 2.

Table 3 :
Numerical solutions for Example 3.

Table 4 :
Numerical solutions for Example 4.

Table 5 :
Numerical solutions for Example 5.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).