Mathematics and Computational Sciences

Chelyshkov polynomials approximation for solving a class of linear and nonlinear equations arising in astrophysics

Document Type : Original Article

Author

Reza Dehghan

Department of Mathematics, Masjed-Soleiman Branch, Islamic Azad University(I.A.U) ,Masjed-Soleiman, Iran

https://doi.org/10.30511/mcs.2024.2020655.1151
Abstract
The aim of this paper is to present a numerical approach for solving linear and nonlinear differential equations arising in astrophysics, commonly known as Lane-Emden equations. The proposed method is based on the collocation method and first involves taking the truncated Chelyshkov series of the function in the equation. The computational cost is reduced due to the orthogonality of Chelyshkov polynomials, and the solution of a linear or nonlinear Lane-Emden equation is reduced to solving a system of linear or nonlinear algebraic equations. Several test examples of these types of differential equations, modeling different physical problems with initial and boundary conditions, are solved to demonstrate the reliability of the method. To demonstrate its effectiveness, absolute error tables and graphs are presented, and the numerical results are compared with other methods and exact solutions. It is observed that when the exact solution has a polynomial form, the proposed method proves to be highly accurate and effective.

Keywords

Chelyshkov polynomials
Lane-Emden equations
Collocation method
Initial and boundary value problems

Subjects

[1] Aslan, B.B., Gurbuz, B., Sezer, M. (2015). A Taylor matrix-collocation method based on residual error for solving Lane-Emden type differential equations. New Trends in Mathematical Sciences, 3(2), 219-224.
[2] Caglar, H., Caglar, N., Ozerc, M. (2009). B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos, Solitons & Fractals, 39(3), 1232-1237.
[3] Chelyshkov, V.S. (2006). Alternative orthogonal polynomials and quadratures. Electronic Transactions on Numerical Analysis, 25(7), 17—26.
[4] Doha, E.H., Abd-Elhameed, W.M., Youssri, Y.H. (2013). Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type. New Astronomy, 23-24, 113-117.
[5] Duggan, R., Goodman, A. (1986). Pointwise bounds for a nonlinear heat conduction model of the human head. Bull. Math. Biol, 48, 229-236.
[6] Erfanifar, R., Sayevand, K., Ghanbari, N., Esmaeili., H. (2021). A modified Chebyshev ϑ-weighted Crank–Nicolson method for analyzing fractional sub-diffusion equations. Numer Methods Partial Differential Eq, 37(1), 614-625.
[7] Gamel, M.E., Abd-El-Hady, M., El-Azab, M. (2017). Chelyshkov-tau approach for solving Bagley Torvik equation. Applied Mathematics, 8, 1795-1807.
[8] Ghorbani, A., Bakherad, M. (2017). A variational iteration method for solving nonlinear Lane-Emden problems. New Astronomy, 54, 1-6.
[9] Gumgum, S. (2020). Taylor wavelet solution of linear and nonlinear Lane-Emden equations. Applied Numerical Mathematics, 158, 44-53.
[10] Gupta, V.G., Sharma, P. (2009). Solving singular initial value problems of Emden-Fowler and Lane-Emden type. Int. J. Appl. Math. Comput, 1, 170-177.
[11] Gurbuz, B., Sezer, M. (2014). Laguerre polynomial approach for solving Lane-Emden type functional differential equations. Appl. Math. Comput, 242, 255-264.
[12] Hosseini, S.G., Abbasbandy, S. (2015). Solution of Lane-Emden type equations by combination of the spectral method and Adomian decomposition method. Math. Probl. Eng, 2015, 1-10.
[13] Isik, O.R., Sezer, M. (2013). Bernstein series solution of a class of Lane-Emden type equations. Mathematical Problems in Engineering.
[14] Khalique, C.M., Ntsime, P. (2008). Exact solutions of the Lane–Emden-type equation. New Astronomy, 13(7), 476-480.
[15] Mohsenyzadeh, M., Maleknejad, K., Ezzati, R. (2015). A numerical approach for the solution of a class of singular boundary value problems arising in physiology. Adv. Differ. Equ, 231.
[16] Moradi, L., Mohammadi, F., Baleanu, D. (2019). A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets. Journal of Vibration and Control, 25(2), 310-324.
[17] Nasab, A.K., Babolian, E., Pashazadeh Atabakan, Z. (2013). Wavelet analysis method for solving linear and nonlinear singular bound ary value problems. Appl. Math. Model, 37, 5876-5886.
[18] Ozturk, Y., Gulsu, M. (2014). An approximation algorithm for the solution of the Lane–Emden type equations arising in astrophysics and engineering using Hermite polynomials. Comp. Appl. Math, 33, 131–145.
[19] Parand, K., Dehghan, M., Rezaei, A.R., Ghaderi, S.M. (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, 1096-1108.
[20] Parand, K., Khaleqi, S. (2016). The rational Chebyshev of second kind collocation method for solving a class of astrophysics problems. Eur. Phys. J. Plus, 131, 24.
[21] Rahimkhani, P., Ordokhani, Y., Lima, P.M. (2019). An improved composite collocation method for distributed-order fractional differential equations based on fractional Chelyshkov wavelets. Applied Numerical Mathematics, 145, 1-27.
[22] Ravi Kanth, A.S.V., Aruna, K. (2010). He’s variational iteration method for treating nonlinear singular boundary value problems. Comput. Math. Appl, 60, 821-829.
[23] Rismani, A.M., Monfared, H. (2012). Numerical solution of singular IVPs of Lane-Emden type using a modified Legendre-spectral method. Appl. Math. Model, 36, 4830-4836.
[24] Roula, P., Madduri, H., Agarwal, R. (2019). A fast-converging recursive approach for Lane-Emden type initial value problems arising in astrophysics. J. Comput. Appl. Math, 359, 182-195.
[25] Saffar Ardabili, J., Talaei, Y. (2018). Chelyshkov, Collocation method for solving the two-dimensional fredholm–volterra integral equations. Int. J. Appl. Comput. Math, 4, 25 (2018).
[26] Sayevand, K., Arab, H. (2018). An efficient extension of the Chebyshev cardinal functions for differential equations with coordinate derivatives of non-integer order. Computational Methods for Differential Equations, 6(3), 339-352.
[27] Sayevand, K., Pichaghchi, K. (2017). A novel operational matrix method for solving singularly perturbed boundary value problems of fractional multi-order. International Journal of Computer Mathematics , 95(4), 767-796.
[28] shokry Mohamed, D. (2019). Chelyshkov’s collocation method for solving three-dimensional linear fredholm integral equations. MathLAB Journal, 4, 163-171.
[29] Singh, R., Kumar, J. (2014). An efficient numerical technique for the solution of nonlinear singular boundary value problems. Comput. Phys. Commun, 185, 1282-1289.
[30] Singh, R., Garg, H., Guleria, V. (2019). Haar wavelet collocation method for Lane-Emden equations with Dirichlet Neumann and Neumann-Robin boundary conditions. J. Comput. Appl. Math, 346,150-161.
[31] Talaei, Y., Asgari, M. (2018). An operational matrix based on Chelyshkov polynomials for solving multi-order fractional differential equations. Neural Computing and Applications, 30, 1369-1376.
[32] Tohidi, E., Erfani, Kh., Gachpazan, M., Shateyi, S. (2013). A New Tau Method for Solving Nonlinear Lane-Emden Type Equations via Bernoulli Operational Matrix of Differentiation. Journal of Applied Mathematics.
[33] Wazwaz, A.M. (2011). The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models. Comm. Nonlinear. Sci. Numer. Simulat, 16, 3881-3886.
[34] Wazwaz, A.M. (2001). A new algorithm for solving differential equations of Lane–Emden type, Appl. Math. Comput, 118 , 287–310.
[35] Yuzbasi, S., Sahin, N. (2011). Numerical solution of a class of Lane-Emden type differential-difference equations. Journal of Advanced Research in Scientific Computing, 3, 14-29.
[36] Yuzbasi, S., Sezer, M. (2013). An improved Bessel collocation method with a residual error function to solve a class of Lane-Emden differential equations. Math. Comput. Model, 57, 1298-1311.
[37] Zhou, F., Xu, X. (2016). Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets. Adv Differ Equ, 17.
Mathematics and Computational Sciences
Volume 5, Issue 3
Summer 2024
Pages 19-37

  • Receive Date 17 January 2024
  • Revise Date 15 September 2024
  • Accept Date 04 October 2024

Home

Submit Manuscript

Reviewers

Contact Us