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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 479396, 9 pages
http://dx.doi.org/10.1155/2011/479396
Research Article

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

1Department of Mathematics, Faculty of Art and Science, Erzincan University, 24100 Erzincan, Turkey
2Department of Mathematics, Young Researchers Club, Azad University, 55518/134 Bonab, Iran
3Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkey

Received 13 January 2011; Accepted 22 February 2011

Academic Editor: Ibrahim Yalcinkaya

Copyright © 2011 Muhammed Yiğider 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

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.

1. Introduction

Lane-Emden equations have the following form [14]: where and are constants, is a continuous real-valued function, and . 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 [911].

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 function is defined as follows: In (2.1), is the original function and is the transformed function, which is called the T-function. Differential inverse transform of 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.

tab1
Table 1: The fundamental operations of one-dimensional differential transform method.

In actual applications, the function is expressed by a finite series and (2.2) can be written as Equation (2.3) implies that is negligibly small. In fact, is decided by the convergence of natural frequency in this study.

Definition 2.1. If , , , and denote convolution. Then .

3. Padé Approximation

Suppose that we are given a power series , representing a function , so that A Padé approximation is a rational fraction which has a Maclaurin expansion which agrees with (3.1) as much as possible. Notice that in (3.2), there are numerator coefficients and denominator coefficients. There is a more or less irrelevant common factor between them, and for definiteness, we take . This choice turns out to be an essential part of the precise definition, and (3.2) is our conventional notation with this choice for . So, there are independent numerator coefficients and independent numerator coefficients, making unknown coefficients in all. This number suggests that normally the ought to fit the power series (3.1) through the orders 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 . And substitute into (3.4), then we have . Therefore, we have constructed a Padé approximation, which agrees with through order . If , where and 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 Table 2 and Figure 1). where the exact solution is . Multiplying both sides of (4.1) by , By using the fundamental operations of one-dimensional differential transform method in Table 1, we obtained following recurrence relation: Then, Power series can be transformed into Padé series

tab2
Table 2: Numerical solution of .
479396.fig.001
Figure 1: Values of and its Padé approximant.

Example 4.2 (see Table 3 and Figure 2). Consider the Lane-Emden-type equation with initial values where the exact solution is .
Multiplying both sides of equation by , Also, and are defined as follows: Then, By using the fundamental operations of one-dimensional differential transform method in Table 1, we obtain the following recurrence relation: Then, Power series can be transformed into Padé series

tab3
Table 3: Numerical solution of .
479396.fig.002
Figure 2: Values of and its 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.

References

  1. A. Aslanov, “Determination of convergence intervals of the series solutions of Emden-Fowler equations using polytropes and isothermal spheres,” Physics Letters A, vol. 372, no. 20, pp. 3555–3561, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. A. Aslanov, “Approximate solutions of Emden-Fowler type equations,” International Journal of Computer Mathematics, vol. 86, no. 5, pp. 807–826, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. E. Momoniat and C. Harley, “Approximate implicit solution of Lane-Emden equation,” New Astronomy, vol. 11, pp. 520–526, 2006. View at Google Scholar
  4. S. A. Yousefi, “Legendre wavelets method for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1417–1422, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. O. U. Richardson, The Emission of Electricity from Hot Bodies, Longmans Green and Company, London, UK, 1921.
  6. A.-M. Wazwaz, “A new algorithm for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 287–310, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Dehghan and F. Shakeri, “Approximate solution of a differential equation arising in astrophysics using the variational iteration method,” New Astronomy, vol. 13, no. 1, pp. 53–59, 2008. View at Publisher · View at Google Scholar
  8. A. Yıldırım and T. Öziş, “Solutions of singular IVPs of Lane-Emden type by the variational iteration method,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 6, pp. 2480–2484, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. I. Ramos, “Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method,” Chaos, Solitons and Fractals, vol. 38, no. 2, pp. 400–408, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. S. K. Vanani and A. Aminataei, “On the numerical solution of differential equations of Lane-Emden type,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2815–2820, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. A. Yıldırım and T. Öziş, “Solutions of singular IVPs of Lane-Emden type by homotopy perturbation method,” Physics Letters. Section A, vol. 369, no. 1-2, pp. 70–76, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. J. K. Zhou, Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.
  13. E. Çelik and M. Bayram, “Arbitrary order numerical method for solving differential-algebraic equation by Padé series,” Applied Mathematics and Computation, vol. 137, no. 1, pp. 57–65, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet