Journal of Advances in Applied Mathematics

Download PDF (611.5 KB) PP. 29 - 43 Pub. Date: January 1, 2016

DOI: 10.22606/jaam.2016.11004

• Xiaoyang Zheng^*
  College of Mathematics and Statistics, Chongqing University of Technology, 400054, Chongqing, China
• Zhengyuan Wei
  College of Mathematics and Statistics, Chongqing University of Technology, 400054, Chongqing, China
• Jiangping He
  College of Mathematics and Statistics, Chongqing University of Technology, 400054, Chongqing, China

This paper presents a new numerical approach, i.e., discontinuous Legendre wavelet Galerkin (DLWG) technique, to solve the Lane-Emden equations. This scheme incorporates Legendre wavelet into discontinuous Galerkin (DG) method, thus it has the advantages of both wavelet Galerkin (WG) method and DG technique. Specifically, the variational formulation of the equation and numerical fluxes are first derived and clearly calculated, then the Lane-Emden equations are converted into solutions to systems of equations. It is pointed that the proposed approach needs less storage space and execution time than the other methods because of the use of discontinuous elements producing lower dimensional, block-diagonal and sparse mass matrices. Furthermore, numerical experiments demonstrate the efficiency and applicability of this technique.

Legendre wavelet, Discontinuous Galerkin method, Lane-Emden equation, Discontinuous Legendre wavelet Galerkin method.

[1] Davis, H.T. Introduction to Nonlinear Differential and Integral Equations. Dover, New York (1962)

[2] Chandrasekhar, S. Introduction to Study of Stellar Structure. Dover, New York (1967)

[3] N.T. Shawagfeh, N.T. “Nonperturbative approximate solution for Lane–Emden equation.” J. Math. Phys. 34 (9), 4364-4369 (1993)

[4] Singh, O.P., Pandey, R.K., Singh, V.K. “An analytic algorithm for Lane–Emden equations arising in Astrophysics using MHAM. Comput.” Phys. Commun. 180, 1116-1124 (2009)

[5] Wazwaz, A.M. “A new algorithm for solving differential equation Lane–Emden type.” Appl. Math. Comput. 118, 287-310 (2001)

[6] Liao, S.J. “A new analytic algorithm of Lane–Emden type equations.” Appl. Math. Comput. 142, 1-16 (2003)

[7] Iqbal, S. and A. Javed, “Application of optimal homotopy asymptotic method for the analytic solution of singular Lane–Emden type equation.” Appl. Math. Comput. 217, 7753--7761 (2011)

[8] He, J.H. “Variational approach to the Lane–Emden equation.” Appl. Math. Comput. 143, 539-541 (2003)

[9] Dehghan, M.F., Shakeri, F. “Approximate solution of a differential equation arising in astrophysics using variational iteration method.” New Astron. 13, 53-59 (2008)

[10] Parand, K., Shahini, M., Dehghan, M. “Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type.” J. Comput. Phys. 228, 8830-8840 (2009)

[11] Van Gorder, R.A. “An elegant perturbation solution for the Lane–Emden equation of the second kind.” New Astron. 16 , 65-67 (2011).

[12] Van Gorder, R.A. “Analytical solutions to a quasilinear differential equation related to the Lane–Emden equation of the second kind. Celestial Mech.” Dynam. Astron. 109, 137-145 (2011)

[13] Razzaghi, M., Yousefi, S. “The Legendre wavelets Operational Matrix of Integration,” International Journal of Systems Science. 32, 500-502 (2001)

[14] Yousefi, S. “Legendre wavelets method for solving differential equations of Lane-Emden type,” Appl. Math. Comput. 181, 1417-1422 (2006)

[15] Razzaghi, M., Yousefi, S. “Legendre wavelets method for the solution of nonlinear problems in the calculus of variations.” Math. Comput. Model. 34, 45-54 (2001)

[16] Mohammadi, F., Hosseini, M.M. “A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations.” Journal of the Franklin Institute. 348, 1787-1796 (2011)

[17] Pandey, Rajesh K., Kumar, Narayan. “Abhinav Bhardwaj and Goutam Dutta, Solution of Lane–Emden type equations using Legendre operational matrix of differentiation.” Applied Mathematics and Computation. 218, 7629-7637 (2012)

[18] Mohammadi, F.M., Hosseini, M.M, Mohyud-din, S.T. “Legendre wavelet Galerkin method for solving ordinary differential equations with nonanalytic solution.” Int. J. Syst. Sci. 42 (4), 579-585 (2011)

[19] Kumar, S., Sloan, I.H. “A new collection-type method for Hammerstein integral equations.” J.Math.Comput. 48, 125-129 (1987)

[20] Alpert, B., Beylkin, G., Gines, D., Vozovoi, L. “Adaptive Solution Partial Differential Equations in Multiwavelet Bases.” J.Comp.Phys. 182, 149-190 (2002)

[21] Wazwaz, A.M. “A new method for solving singular initial value problems in the second-order ordinary differential equations.” Appl. Math. Comput. 128, 45-57 (2002)

[22] Avudainay agarn, A., Vano, C. “Wavelet-Galerkin method for integro-differential equations.” Appl.Numer.Math. 32, 247-254 (2000)

[23] Mohammadi, F., Hosseini, M.M. “Legendre wavelet method for solving linear stiff systems.” J. Adv. Res. Differential Equations. 2 (1), 47--57 (2010)

[24] Zheng Xiaoyang, Yang Xiaofan, Su Hong, Qiu, Liqiong “Discontinuous Legendre wavelet element method for elliptic partial differential equations.” Applied Mathematics and Computation. 218, 3002-3018 (2011)

[25] Vidden Chad, Yan, Jue “A new direct discontinuous Galerkin method with symmetric structure for nonlinear diffusion equations.” Journal of Computational Mathematics. Vol.31, No.6, 638-662 (2013)