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

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  • Ahmad Sami Bataineh

    (Department of Mathematics, Faculty of Science, Al-Balqa’ Applied University, Al Salt 19117, Jordan
    Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, Bangi 43600 UKM, Selangor, Malaysia)

  • 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)

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.

Suggested Citation

  • Ahmad Sami Bataineh & Osman Rasit Isik & Abedel-Karrem Alomari & Mohammad Shatnawi & Ishak Hashim, 2020. "An Efficient Scheme for Time-Dependent Emden-Fowler Type Equations Based on Two-Dimensional Bernstein Polynomials," Mathematics, MDPI, vol. 8(9), pages 1-17, September.
  • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1473-:d:407124
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    References listed on IDEAS

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    1. Ramos, J.I., 2008. "Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 400-408.
    2. He, Ji-Huan, 2005. "Application of homotopy perturbation method to nonlinear wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 695-700.
    3. Kashkari, Bothayna S.H. & Syam, Muhammed I., 2016. "Fractional-order Legendre operational matrix of fractional integration for solving the Riccati equation with fractional order," Applied Mathematics and Computation, Elsevier, vol. 290(C), pages 281-291.
    4. Asgari, M. & Ezzati, R., 2017. "Using operational matrix of two-dimensional Bernstein polynomials for solving two-dimensional integral equations of fractional order," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 290-298.
    5. Osman Rasit Isik & Mehmet Sezer, 2013. "Bernstein Series Solution of a Class of Lane-Emden Type Equations," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-9, April.
    Full references (including those not matched with items on IDEAS)

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