The Numerical Method for Solving Differential Equations of Lane-Emden Type by Padé Approximation

Numerical solution differential equation of Lane-Emden type is considered by Padé approximation. We apply these method to two examples. First differential equation of Lane-Emden type has been converted to power series by one-dimensional differential transformation, then the numerical solution of equation was put into Padé series form. Thus, we have obtained numerical solution differential equation of Lane-Emden type.


Introduction
Lane-Emden equations have the following form 1-4 : where A and B are constants, f x, y is a continuous real-valued function, and g x ∈ C 0, 1 .Lane-Emden equations are singular initial value problems relating to second-order ordinary differential equations ODEs which have been used to model several phenomena in mathematical physics and astrophysics such as thermal explosions 5 , the thermal behavior of a spherical cloud of gas, isothermal gas spheres, and thermionic currents 5 .Several methods for the solutions of Lane-Emden equations have been presented.Wazwaz 6 has given a general way to construct exact and series solutions to Lane-Emden equations by employing the Adomian decomposition method.A numerical solution of Lane-Emden equations is given based on the Legendre wavelets methods 4 .The variational iteration method is used to solve differential equations arising in astrophysics including the Lane-Emden equation 7, 8 .The homotopy perturbation method and the integral operator method were suggested in 9-11 .
In this paper, we apply these method to two examples.First, differential equation of Lane-Emden type has been converted to power series by one-dimensional differential transformation; differential transformation was introduced first by Zhou 12 .Then the numerical solution of equation was put into Padé series form 13 .The Padé approximation method was used to accelerate the convergence of the power series solution.Thus, we obtain numerical solution differential equation of Lane-Emden type.

One-Dimensional Differential Transform
Differential transform of function y x is defined as follows: In 2.1 , y x is the original function and Y k is the transformed function, which is called the T-function.Differential inverse transform of Y k is defined as From 2.1 and 2.2 , we obtain Equation 2.3 implies that the concept of differential transform is derived from Taylor series expansion, but the method does not evaluate the derivatives symbolically.However, relative derivatives are calculated by an iterative way which is described by the transformed equations of the original functions.In this study, we use the lower case letter to represent the original function, and upper case letter represent the transformed function.
From the definitions of 2.1 and 2.2 , it is easily proven that the transformed functions comply with the basic mathematics operations shown in Table 1.

Table 1:
The fundamental operations of one-dimensional differential transform method.

Original function
Transformed function In actual applications, the function y x is expressed by a finite series and 2.2 can be written as In fact, m is decided by the convergence of natural frequency in this study.

Pad é Approximation
Suppose that we are given a power series ∞ i 0 a i x i , representing a function f x , so that which has a Maclaurin expansion which agrees with 3.1 as much as possible.Notice that in 3.2 , there are L 1 numerator coefficients and M 1 denominator coefficients.There is a more or less irrelevant common factor between them, and for definiteness, we take q 0 1.This choice turns out to be an essential part of the precise definition, and 3.2 is our conventional notation with this choice for q 0 .So, there are L 1 independent numerator coefficients and M independent numerator coefficients, making L M 1 unknown coefficients in all.This number suggests that normally the L/M ought to fit the power series 3.1 through the orders 1, x, x 2 , . . ., x L M in the notation of formal power series Multiply both sides of 3.3 by the denominator of right side in 3.3 , and compare the coefficients of both sides of 3.3 , then we have Solve the linear equation in 3.5 , then we have q k k 1, . . ., L .And substitute q k into 3.4 , then we have p 1 l 0, . . ., M .Therefore, we have constructed a L/M Padé approximation, which agrees with ∞ i 0 a i x i through order x L M .If M ≤ L ≤ M 2, where M and L are the degree of numerator and denominator in Padé series, respectively, then Padé series gives an A-stable formula for an ordinary differential equation.

Numerical Examples
Example 4.1 see Table 2 and Figure 1 .
where the exact solution is u x e x 2 .Multiplying both sides of 4.1 by x, By using the fundamental operations of one-dimensional differential transform method in Table 1, we obtained following recurrence relation:  Then,

4.4
Power series u * x can be transformed into Padé series  1, we obtain the following recurrence relation:  Then,

Conclusion
A Padé approximation method has proposed for solving differential equation of Lane-Emden type in this study.This method is very simple and effective for most of Lane-Emden equations.Numerical results explicitly reveal the complete reliability of the proposed Padé approximation.

Table 2 :
Numerical solution of u x .

Table 3 :
Numerical solution of y x .