An efficient high-order compact finite difference scheme for Lane-Emden-type equations

Document Type : Research Paper

Authors

Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.

Abstract

In this paper, an efficient high-order compact finite difference (HOCFD) scheme is introduced for solving generalized Lane-Emden equations. For nonlinear types, it is shown that a combined quasilinearization and HOCFD scheme gives excellent results, while a few quasilinear iterations are needed. Then the proposed method is developed for solving the system of linear and nonlinear Lane-Emden equations. Some numerical examples are provided, and the obtained results of the proposed method are then compared with previous well-established methods. The numerical experiments show the accuracy and efficiency of the proposed method.

Keywords

Main Subjects

References

  • [1] B. Azarnavid, F. Parvaneh, and S. Abbasbandy, Picard-reproducing kernel Hilbert space method for solving generalized singular nonlinear Lane-Emden type equations, Mathematical Modelling and Analysis, 20(6) (2015), 754–767.
  • [2] R. E. Bellman and R. K. Kalaba, Nonlinear Boundary-Value Problems, American Elsevier Publishing Co., Inc., New York, 1965.
  • [3] A. H. Bhrawy and A. S. Alofi, A Jacobi–Gauss collocation method for solving nonlinear Lane–Emden type equations, Communications in Nonlinear Science and Numerical Simulation, 17(1) (2012), 62–70.
  • [4] M. Bisheh Niasar, A Computational Method for Solving the Lane-Emden Initial Value Problems, Computational Methods for Differential Equations, 8(4) (2020), 673-684.
  • [5] S. D. Conte and C. De Boor, Elementary Numerical Analysis: an Algorithmic Approach, Society for Industrial and Applied Mathematics, 2017.
  • [6] H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, US Government Printing Office, 1961.
  • [7] R. Doostaki, M. M. Hosseini, and A. Salemi, A new simultaneously compact finite difference scheme for highdimensional time-dependent PDEs, Mathematics and Computers in Simulation, 212 (2023), 504-523.
  • [8] Y. Q. Hasan and L. M. Zhu, Solving singular boundary value problems of higher-order ordinary differential equations by modified Adomian decomposition method, Communications in Nonlinear Science and Numerical Simulation, 14(6) (2009), 2592–2596.
  • [9] G. P. Horedt, Polytropes: applications in astrophysics and related fields, Springer Science and Business Media, 2004.
  • [10] M. Izadi, A discontinuous finite element approximation to singular Lane-Emden type equations, Applied Mathematics and Computation, 401 (2021), 126115.
  • [11] R. E. Kalaba, On Nonlinear Differential Equations, the Maximum Operation, and Monotone Convergence, New York University, 1957.
  • [12] A. R. Kanth and K. Aruna, Solution of singular two-point boundary value problems using differential transformation method, Physics Letters A, 372(26) (2008), 4671–4673.
  • [13] M. Lakestani and M. Dehghan, Four techniques based on the B-spline expansion and the collocation approach for the numerical solution of the Lane-Emden equation, Mathematical Methods in the Applied Sciences, 36(16) (2013), 2243–2253.
  • [14] S. K. Lele, Compact finite difference schemes with spectral-like resolution, Journal of computational physics, 103(1) (1992), 16–42.
  • [15] S. Mall and S. Chakraverty, Chebyshev neural network based model for solving Lane-Emden type equations, Applied Mathematics and Computation, 247 (2014), 100–114.
  • [16] V. Mandelzweig and F. Tabakin, Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs, Computer Physics Communications, 141(2) (2001), 268–281.
  • [17] M. P. Mehra and K. S. Patel, Algorithm 986: a suite of compact finite difference schemes, ACM Transactions on Mathematical Software, 44(2) (2017), 1–31.
  • [18] R. Mohammadzadeh, M. Lakestani, and M. Dehghan, Collocation method for the numerical solutions of Lane–Emden type equations using cubic Hermite spline functions, Mathematical Methods in the Applied Sciences, 37(9) (2014), 1303-1717.
  • [19] Y. Oztürk, solution for the system of Lane–Emden type equations using Chebyshev polynomials, Mathematics, 5(10) (2018), 181.
  • [20] R. K. Pandey, N. Kumar, and A. Bhardwaj, Solution of Lane–Emden type equations using Legendre operational matrix of differentiation, Applied Mathematics and Computation, 21814 (2012), 7629–7637.
  • [21] R. K. Pandey and N. Kumar, Solution of Lane–Emden type equations using Bernstein operational matrix of differentiation, New Astronomy, 17(3) (2012), 303–308.
  • [22] K. Parand and M. Delkhosh, An effective numerical method for solving the nonlinear singular Lane-Emden type equations of various orders, Jurnal Teknologi, 79(1) (2017).
  • [23] K. Parand, M. Dehghan, A. Rezaei, and S. Ghaderi, 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) (2010), 1096–1108.
  • [24] K. Parand, M. Shahini, and M. Dehghan, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type, Journal of Computational Physics, 228(23) (2009), 8830–8840.
  • [25] J. Ramos, Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method, Chaos, Solitons & Fractals, 38(2) (2008), 400–408.
  • [26] J. I. Ramos, Linearization techniques for singular initial-value problems of ordinary differential equations, Applied Mathematics and Computation, 161(2) (2005), 525–542.
  • [27] P. Roul, K. Thula, and V. P. Goura, An optimal sixth-order quartic B-spline collocation method for solving Bratutype and Lane-Emden type problems, Mathematical Methods in the Applied Sciences, 42(8) (2019), 2613–2630.
  • [28] S. Shiralashetti and S. Kumbinarasaiah, Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane-Emden type equations, Applied Mathematics and computation, 315 (2017), 591–602.
  • [29] O. P. Singh, R. K. Pandey, and V. K. Singh, An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified homotopy analysis method, Computer Physics Communications, 180(7) (2009), 1116– 1124.
  • [30] A. M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Applied mathematics and computation, 118(2-3) (2001), 287–310.
  • [31] A. M. Wazwaz, A new method for solving singular initial value problems in the second-order ordinary differential equations, Applied Mathematics and computation, 128(1) (2002), 45–57.
  • [32] A. M. Wazwaz, R. Rach, and J. S. Duan, A study on the systems of the Volterra integral forms of the Lane-Emden equations by the Adomian decomposition method, Mathematical Methods in the Applied Sciences, 37(1) (2014), 10–19.
  • [33] S. A. Yousefi, Legendre wavelets method for solving differential equations of Lane-Emden type, Applied Mathematics and Computation, 181(2) (2006), 1417–1422.
  • [34] A. Yildirim, T. Ozis, Solutions of singular IVPs of Lane-Emden type by the variational iteration method, Nonlinear Analysis: Theory, Methods & Applications, 70(6) (2009), 2480–2484.
  • [35] A. Zamiri, A. Borhanifar, and A. Ghannadiasl, Laguerre collocation method for solving Lane-Emden type equations, Computational Methods for Differential Equations, 9(4) (2021), 1176-1197.
Computational Methods for Differential Equations
Volume 13, Issue 1
January 2025
Pages 107-122

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History

  • Receive Date: 21 April 2023
  • Revise Date: 22 November 2023
  • Accept Date: 26 December 2023

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