Document Type : Research Paper

Authors

1 Department of Applied Mathematics, School of Mathematical Sciences, University of Guilan, Rasht, Iran.

2 Department of Applied Mathematics, School of Mathematical Sciences, University of Guilan, University Campus2, Rasht, Iran.

Abstract

In this paper we construct Non-Standard finite difference schemes (NSFD) for numerical solution of nonlinear Lane-Emden type equations which are nonlinear ordinary dierential equations on semi-infinite domain. They are categorized as singular initial value problems. This equation describes a variety of phenomena in theoretical physics and astrophysics.  The presented schemes are obtained by using the Non-Standard finite difference method. The use of NSFD method and its approximations play an important role for the formation of stable numerical methods. The main advantage of the schemes is that the algorithm is very simple and very easy to implement. Thus, this method may be applied as a simple and accurate solver for ODEs and PDEs and it can also be utilized as an accurate algorithm to solve linear and nonlinear equations arising in physics and other fields of applied mathematics. Illustrative examples have been discussed to demonstrate validity and applicability of the technique and the results have been compared with the exact solutions.

Keywords

Main Subjects

[1]   H. Aminikhah, S. Moradian, (2013). Numerical Solution of Singular Lane-Emden Equation. Hindawi Publishing Corporation ISRN Mathematical Physics, 2013, 1-9.
 
 
[2]    S. Liao, (2003). a new analytic algorithm of Lane-Emden type equations. Applied Mathematics and Computation, 142 (1), 1–16.
[3]   J. H. He, (2003). Variational approach to the Lane-Emden equation. Applied Mathematics and Computation, 143 (2), 539–541.
[4]   C. M. Bender, K. A. Milton, S. S. Pinsky, and L. M. Simmons, Jr., (1989). A new perturbative approach to nonlinear problems. Journal of Mathematical Physics, Vol. 30 (7), 1447–1455.
[5]    N. T. Shawagfeh, (1993). Non-perturbative approximate solution for Lane-Emden equation. Journal of Mathematical Physics, 34 (9), 4364–4369.
[6]   A. M. Wazwaz, (2001). a new algorithm for solving differential equations of Lane-Emden type. Applied Mathematics and Computation, 118 (2), 287–310.
[7]    A. M. Wazwaz, (2002). a new method for solving singular initial value problems in the second-order ordinary differential equations. Applied Mathematics and Computation, 128 (1), 45–57.
[8]    J. I. Ramos, (2008). Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method. Chaos, Solitons and Fractals, 38 (2), 400–408.
[9]    K. Parand and M. Razzaghi, (2004). Rational Chebyshev tau method for solving higher-order ordinary differential equations. International Journal of Computer Mathematics, Vol. 81 (1), 73–80.
[10]  K. Parand and M. Razzaghi, (2004). Rational Legendre approximation for solving some physical problems on semi-infinite intervals. Physica Scripta, 69, 353–357.
[11]  K. Parand, M. Shahini, and M. Dehghan, (2009). Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type. Journal of Computational Physics, 228 (23), 8830– 8840.
[12]   K. Parand, M. Dehghan, A. R. Rezaei, and S. M. Ghaderi, (2010). an approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method. Computer Physics Communications, 181 (6), 1096–1108.
[13]  M. El-Gebeily and D. O’Regan, (2007). a quasilinearization method for a class of second order singular nonlinear differential equations with nonlinear boundary conditions. Nonlinear Analysis: Real World Applications, 8, 174–186.
[14]  V. B. Mandelzweig and F. Tabakin, (2001). Quasi linearization approach to nonlinear problems in physics with application to nonlinear ODEs. Computer Physics Communications, 141 (2), 268–281.
[15]    J. I. Ramos, (2003). Linearization methods in classical and quantum mechanics. Computer Physics Communications, 153 (2), 199–208.
[16]   Y. Bozhkov and A. C. Gilli Martins, (2004). Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents. Nonlinear Analysis: Theory, Methods & Applications, 57 (5), 773–793.
[17]   E. Momoniat and C. Harley, (2006). Approximate implicit solution of a Lane-Emden equation. New Astronomy, 11, 520–526.
[18]    T. Özis and A. Yildirim, (2007). Solutions of singular IVP’s of Lane-Emden type by homotopy pertutbation method. Physics Letters A, 369, 70–76.
[19]  T. Özis and A. Yildirim, (2009). Solutions of singular IVPs of Lane-Emden type by the variational iteration method. Nonlinear Analysis: Theory, Methods & Applications, 70, 6, 2480–2484.
[20]   Ronald E. Mickens, (2001). A Non-Standard Finite Difference Scheme for a Nonlinear PDE Having Diffusive Shock Wave Solutions. Mathematics and Computers in Simulation, 55, 549-555.
[21]  Ronald E. Mickens, (2003). A Non-Standard Finite Difference Scheme for a Fisher PDE Having Nonlinear Diffusion, Diffusion. Comput. Math. App, 145, 429-436.
[22]  Ronald E. Mickens, (2005). A Non-Standard Finite Difference Scheme for a PDE Modeling Combustion with Nonlinear Advection and Diffusion. Mathematics and Computers in Simulation, 69, 439-446.
[23]   Ronald E. Mickens, (2005). Advances in the Applications of Non-Standard Finite Difference Schemes. World Scientific, Singapore.
[24]   Ronald E. Mickens, (2006). Calculation of Denominator Functions for Non-Standard Finite Difference Schemes for Differential Equations Satisfying a Positivity Condition. Wiley Inter Science, 23, 672-628.
[25]  Ronald E. Mickens, (2007). Determination of Denominator Functions for a NSFD Scheme for the Fisher PDE with Linear Advection. Mathematics and Computers in Simulation, 74, 127-195.
[26]   Alvarez-Ramirez, J. Valdes, (2009). Non-Standard Finite Differences schemes for Generalized Reaction- Diffusion Equations. Computational and Applied Mathematics, 228, 334-343.
 
 
 
 
[27]    Benito M. Chen-Charpentier, Dobromir T. Dimitrov, Hristo V. Kojouharov, (2006). Combined Non- Standard Numerical Methods for ODEs with Polynomial Right-Hand Sides. Mathematics and Computers in Simulation,73, 105-113.
[28]  Elizeo Hernandez-Martinez, Francisco J. Valdes-Prada, Jose Alvarez-Ramirez, (2011). A Greens Function Formulation of Nonlocal Finite-Difference Schemes for Reaction-Diffusion Equations. Computational and Applied Mathematics, 235, 3096-3103.
[29]   K. Moaddy, S. Momani, I. Hashim, (2011). The Non-Standard Finite Difference Scheme for linear Fractional PDEs in Fluid Mechanics. Computers and Mathematics with Applications, 61, 1209-1216.