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

Authors: Muhammed Yiğider; Khatereh Tabatabaei; Ercan Çelik
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

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.


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## Body

## 1. Introduction

Lane-Emden equations have the following form [1–4]:(1.1)y′′+kxy′+f(x,y)=g(x),0≤x≤1,k≥0,y(0)=A,y′(0)=B,
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.


## 2. One-Dimensional Differential Transform

Differential transform of functiony(x) is defined as follows:(2.1)Y(k)=1k![dky(x)dxk]x=0.
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(2.2)y(x)=∑k=0∞xkY(k).
From (2.1) and (2.2), we obtain(2.3)y(x)=∑k=0∞xkk![dky(x)dxk]x=0.
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 functionTransformed functiony(x)=u(x)±v(x)Y(k)=U(k)±V(k)y(x)=cw(x)Y(k)=cW(k)y(x)=dw/dxY(k)=(k+1)W(k+1)y(x)=djw/dxjY(k)=(k+1)(k+2)⋯(k+j)W(k+j)y(x)=u(x)v(x)Y(k)=∑r=0kU(r)V(k-r)y(x)=xjY(k)=δ(k-j)={1,k=j0,k≠jIn actual applications, the functiony(x) is expressed by a finite series and (2.2) can be written as  (2.4)y(x)=∑k=0mxkY(k).
Equation (2.3) implies that ∑k=m+1∞xkY(k) is negligibly small. In fact, m is decided by the convergence of natural frequency in this study.Definition 2.1.
Ify(x)=u(x)v(x), u(x)=D-1[U(k)], v(x)=D-1[V(k)], and ⊗ denote convolution. Then [y(x)]=D[u(x)v(x)]=U(k)⊗V(k)=∑r=0kU(r)V(k-r).


## 3. Padé Approximation

Suppose that we are given a power series∑i=0∞aixi, representing a function f(x), so that(3.1)f(x)=∑i=0∞aixi.
A Padé approximation is a rational fraction(3.2)[LM]=p0+p1x+⋯+pLxLq0+q1x+⋯+qMxM,
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 q0=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 q0. 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,x2,…,xL+M in the notation of formal power series(3.3)∑i=0∞aixi=p0+p1x+⋯+pLxLq0+q1x+⋯+qMxM+O(xL+M+1).
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 (3.4)al+∑k=1mal-kqk=pl,(l=0,…,M),(3.5)al+∑k=1Lal-kqk=0,(l=M+1,…,M+L).
Solve the linear equation in (3.5), then we have qk(k=1,…,L). And substitute qk into (3.4), then we have p1(l=0,…,M). Therefore, we  have constructed a [L/M] Padé approximation, which agrees with ∑i=0∞aixi through order xL+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.


## 4. Numerical Examples

Example 4.1 (see Table2 and Figure 1).
(4.1)u′′(x)+2xu′(x)=2(2x2+3)u(x),0≤x≤1,u(0)=1,u′(0)=0,
where the exact solution is u(x)=ex2. Multiplying both sides of (4.1) by x,(4.2)xu′′(x)+2u′(x)=4x3u(x)+6xu(x),0≤x≤1.
By using the fundamental operations of one-dimensional differential transform method in Table 1,  we obtained following recurrence relation:
(4.3)δ(k-1)⊗(k+1)(k+2)U(k+2)+2(k+1)U(k+1)=4δ(k-3)⊗U(k)+6δ(k-1)⊗U(k).
Then,
(4.4)∑l=0kδ(l-1)(k-l+1)(k-l+2)U(k-l+2)+2(k+1)U(k+1)=4∑l=0kδ(l-3)U(k-l)+∑l=0k6δ(l-1)U(k-l),U(0)=1,U(1)=0,U(2)=1,U(3)=0,U(4)=12,U(5)=0,U(6)=16,U(7)=0,U(8)=124,U(9)=0,…,u*(x)=U(0)+xU(1)+x2U(2)+x3U(3)+…,u*(x)=1+x2+12x4+16x6+124x8+⋯=1+x2+12x4+13!x6+14!x8+⋯+1n!x2n.
Power series u*(x) can be transformed into Padé series
(4.5)P[54]=(1+(1/2)x2+(1/12)x4)(1-(1/2)x2+(1/12)x4)=(1+0.5x2+0.8333333333x4)(1-0.5x2+0.8333333333x4).Table 2
Numerical solution ofu(x).
xu(x)P[5/4]|u(x)-P[5/4]|0.11.0100501671.01005016700.21.0408107741.04081077400.31.0941742341.0941742754.1×10-80.41.1735099741.1735107007.26×10-70.51.2840169271.2840236680.0000067410.61.4332758401.433172850.0000414450.71.6320601671.6322512660.0001910990.81.8954811741.8961911870.0007100130.92.2445596342.2467772350.0022176011.02.7083333342.7142857140.005952380Figure 1
Values ofu(x) and its P[5/4] Padé approximant.Example 4.2 (see Table3 and Figure 2).
Consider the Lane-Emden-type equation(4.6)y′′+2xy′+(8ey+4ey/2)=0,
with initial values
(4.7)y(0)=0,y′(0)=0,
where the exact solution is y(x)=-2ln(1+x2).
Multiplying both sides of equation byx,(4.8)xy′′+2y′+8xey+4xey/2=0.
Also,  ey and  ey/2 are defined as follows:
(4.9)ey=1+y,ey/2=1+y2.
Then,
(4.10)xy′′+2y′+12x+10xy=0.
By using the fundamental operations of one-dimensional differential transform method in Table 1, we obtain the following recurrence relation:
(4.11)δ(k-1)⊗(k+1)(k+2)Y(k+2)+2(k+1)Y(k+1)+12δ(k-1)+10δ(k-1)⊗Y(k)=0.
Then,
(4.12)∑l=0kδ(l-1)(k-l+1)(k-l+2)Y(k-l+2)+2(k+1)Y(k+1)+12δ(k-1)+10∑l=0kδ(l-1)Y(k-l)=0,y(0)=0,y(1)=0,y(2)=-2,y(3)=0,y(4)=1,y(5)=0,y(6)=-521,y(7)=0,y(8)=25756,…,y*(x)=y(0)+xy(1)+x2y(2)+x3y(3)+…,y*(x)=-2x2+x4-521x6+25756x8+….
Power series y*(x)can be transformed into Padé series
(4.13)P[54]=((589/704)x4-2x2)(1+(115/1408)x2-(4625/59136)x4)=(0.8366477273x4-2x2)(1+0.08167613636x2-0.07820955087x4).Table 3
Numerical solution ofy(x).
xy(x)P[5/4]|y(x)-P[5/4]|0.1−0.01990023712−0.019900237131×10-110.2−0.07841498810−0.078414990802.70×10-90.3−0.1720671642−0.17206731621.520×10-70.4−0.2953112381−0.29531386990.00000263180.5−0.4408387684−0.44086250910.00002374070.6−0.5998683214−0.60000979520.00014147380.7−0.7622819782−0.76291414750.00063216930.8−0.9164312381−0.91871602860.00228479050.9−1.048395758−1.0554085360.0070127781.0−1.140438988−1.1593333450.018894357Figure 2
Values ofy(x) and its P[5/4] Padé approximant.


## 5. 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.



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