Laguerre collocation method for solving Lane-Emden type equations

Document Type : Research Paper


1 Department of Mathematics, University of Mohaghegh ardabili, Ardabil, Iran.

2 Department of Civil Engineering, University of Mohaghegh Ardabili, Ardabil, Iran.


In this paper, a Laguerre collocation method is presented in order to obtain numerical solutions for linear and nonlinear Lane-Emden type equations and their initial conditions. The basis of the present method is operational matrices with respect to modified generalized Laguerre polynomials(MGLPs) that transforms the solution of main equation and its initial conditions to the solution of a matrix equation corresponding to the system of algebraic equations with the unknown Laguerre coefficients. By solving this system, coefficients of approximate solution of the main problem will be determined. Implementation of the method is easy and has more accurate results in comparison with results of other methods.


  • [1]          H. Adibi and A. M. Rismani, On using a modified Legendre-spectral method for solving singular IVPs of Lane-Emden type, Comput. Math. Appl., 60 (2010), 2126–2130.
  • [2]          A. Borhanifar and A. Zamiri, Application of ( G′ )-expansion method for the Zhiber-Shabat equa- tion and other related equations, Math. Comput. Model., 54 (2011), 2109–2116.
  • [3]          A. Borhanifar and M. M. Kabir, New periodic and soliton solutions by application of Exp- function method for nonlinear evolution equations, J. Comput. Appl. Math., 229 (2009), 158– 167.
  • [4]          A. Borhanifar and R. Abazari. Numerical study of nonlinear Schrdinger and coupled Schrdinger equations by differential transformation method , Opt. Commun., 283 (2010), 2026–2031.
  • [5]          A. Borhanifar and Kh. Sadri, A new operational approach for numerical solution of generalized functional integro-differential equations, J. Comput. Appl. Math., 279 (2015), 80–96.
  • [6]          A. H. Bhrawy and A. S. Alofi, A JacobiGauss collocation method for solving nonlinear LaneEm- den type equations, Commun. Nonlinear. Sci., 17 (2012), 62–70.
  • [7]          J. P. Boyd, Chebyshev spectral methods and the Lane-Emden problem, Numer. Math. Theor. Meth. Appl., 4 (2011), 142–157.
  • [8]          C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods in fluid dynamics, Springer, New York, 1988.
  • [9]          H. T. Davis, Introduction to nonlinear differential and integral equations, Dover publication, Inc, New York, 1962.
  • [10]        M. Dehghan, M. Shakourifar, and A. Hamidi, The solution of linear and nonlinear systems of Volterra functional equations using AdomianPade technique, Chaos. Soliton. Fract., 39 (2009), 2509–2521.
  • [11]        E. H. Doha, W. M. Abd-Elhameed, and Y. H. Youssri, Second kind Chebyshev operational matrix algorithm for solving differential equations of LaneEmden type, New. Astron., 23 (2013), 113117.
  • [12]        D. Funaro, Polynomial approximation of differential equations, Springer-Verlag, Berlin, 1992.
  • [13]        B. Y. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential on a semi- infinite interval, Numer. Math., 86 (2000), 635–654.
  • [14]        B. Y. Guo and Z. Xiao-yong, A new generalized Laguerre approximation and its applications, J. Comput. Appl. Math., 184 (2005), 382–403.
  • [15]        B. Y. Guo, Spectral methods and their applications, World Scientific, 1998.
  • [16]        K. Parand, M. Dehghan, A. R. Rezaei,  and S.  M.  Ghaderi,  An  approximate algorithm for  the solution of the nonlinear LaneEmden type equations arising in astrophysics using Hermite function collocation method, Comput. Phys. Commun., 181 (2010), 1096–1108.
  • [17]        K. P. Rajesh, N. Kumar, A. Bhardwaj, and G. Dutta, Solution of LaneEmden type equations using Legendre operational matrix of differentiation, Appl. Math. Comput., 218 (2012), 7629– 7637.
  • [18]        O. W. Richardson, The emission of electricity from hot bodies, Longmans, Green and Company, 1921.
  • [19]        J. Shen, T. Tang, and L. L. Wang, Spectral methods: algorithms, analysis and applications, Springer, New York, 2011.
  • [20]        S. C. Shiralashetti and S. Kumbinarasaiah, Hermite wavelets method for the numerical solution of linear and nonlinear singular initial and boundary value problems, Comput. Methods Differ. Equ., 7 (2019), 177–198.
  • [21]        S. C. Shiralashetti and S. Kumbinarasaiah, New generalized operational matrix of integration to solve nonlinear singular boundary value problems using Hermite wavelets, Arab journal of basic and applied sciences, 26 (2019), 385–396.
  • [22]        S. C. Shiralashetti and S. Kumbinarasaiah, Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear LaneEmden type equations, Appl. Math. Comput., 315 (2017), 591–602.
  • [23]        G. Szego, Orthogonal polynomils, Colloquium publications, AMS, New York, 1939.
  • [24]        L. N. Trefethen, Spectral methods in MATLAB, SIAM, Philadelphia, PA, 2000.
  • [25]        Z. Q. Wang, The Laguerre spectral method for solving Neumann boundary value problems, J. Comput. Appl. Math., 235 (2011), 3229–3237.
  • [26]        A. M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Appl. Math. Comput., 118 (2001), 287–310.
  • [27]        A. M. Wazwaz, The modified decomposition method for analytic treatment of differential equa- tions, Appl. Math. Comput., 173 (2006), 165–176.
  • [28]        A. Yildirim and T. Ozis, Solutions of singular IVPs of LaneEmden type by the variational iteration method, Nonlinear. Anal., 70 (2009), 2480–2484.
  • [29]        S. Yzbasi and M. Sezer, An improved Bessel collocation method with a residual error function to solve a class of LaneEmden differential equations, Math. Comput. Model., 57 (2013), 1298– 1311.
  • [30]        H. Zhu, J. Niu, R. Zhang, and Y. Lin, A new approach for solving nonlinear singular boundary value problems, Math. Model. Anal., 23 (2018), 33–43.