An Efficient Scheme for Time-Dependent Emden-Fowler Type Equations Based on Two-Dimensional Bernstein Polynomials

Ahmad Sami Bataineh, Osman Rasit Isik, Abedel-Karrem Alomari, Mohammad Shatnawi and Ishak Hashim
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Ahmad Sami Bataineh: Department of Mathematics, Faculty of Science, Al-Balqa’ Applied University, Al Salt 19117, Jordan
Osman Rasit Isik: Elementary Mathematics Education Program, Faculty of Education, Mugla Sitki Kocman University, Mugla 48000, Turkey
Abedel-Karrem Alomari: Department of Mathematics, Faculty of Science, Yarmouk University, Irbid 21163, Jordan
Mohammad Shatnawi: Department of Basic Science, Al-Huson University College, Al-Balqa’ Applied University, P.O. Box 50, Al-Huson, Irbid 21510, Jordan
Ishak Hashim: Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, Bangi 43600 UKM, Selangor, Malaysia

Mathematics, 2020, vol. 8, issue 9, 1-17

Abstract: In this study, we introduce an efficient computational method to obtain an approximate solution of the time-dependent Emden-Fowler type equations. The method is based on the 2D-Bernstein polynomials (2D-BPs) and their operational matrices. In the cases of time-dependent Lane–Emden type problems and wave-type equations which are the special cases of the problem, the method converts the problem to a linear system of algebraic equations. If the problem has a nonlinear part, the final system is nonlinear. We analyzed the error and give a theorem for the convergence. To estimate the error for the numerical solutions and then obtain more accurate approximate solutions, we give the residual correction procedure for the method. To show the effectiveness of the method, we apply the method to some test examples. The method gives more accurate results whenever increasing n , m for linear problems. For the nonlinear problems, the method also works well. For linear and nonlinear cases, the residual correction procedure estimates the error and yields the corrected approximations that give good approximation results. We compare the results with the results of the methods, the homotopy analysis method, homotopy perturbation method, Adomian decomposition method, and variational iteration method, on the nodes. Numerical results reveal that the method using 2D-BPs is more effective and simple for obtaining approximate solutions of the time-dependent Emden-Fowler type equations and the method presents a good accuracy.

Keywords: Bernstein polynomials; operational matrices; Emden-Fowler equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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