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A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

Author

Listed:
  • Constantin Bota

    (Department of Mathematics, Politehnica University of Timişoara, 300006 Timişoara, Romania
    These authors contributed equally to this work.)

  • Bogdan Căruntu

    (Department of Mathematics, Politehnica University of Timişoara, 300006 Timişoara, Romania
    These authors contributed equally to this work.)

  • Dumitru Ţucu

    (Department of Mechanical Machinery, Equipment and Transport, Politehnica University of Timişoara, 300222 Timişoara, Romania
    These authors contributed equally to this work.)

  • Marioara Lăpădat

    (Department of Mathematics, Politehnica University of Timişoara, 300006 Timişoara, Romania
    These authors contributed equally to this work.)

  • Mădălina Sofia Paşca

    (Department of Mathematics, Politehnica University of Timişoara, 300006 Timişoara, Romania
    Department of Mathematics, West University of Timişoara, 300223 Timişoara, Romania
    These authors contributed equally to this work.)

Abstract

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

Suggested Citation

  • Constantin Bota & Bogdan Căruntu & Dumitru Ţucu & Marioara Lăpădat & Mădălina Sofia Paşca, 2020. "A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order," Mathematics, MDPI, vol. 8(8), pages 1-12, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1336-:d:397262
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    References listed on IDEAS

    as
    1. Ya Qin & Adnan Khan & Izaz Ali & Maysaa Al Qurashi & Hassan Khan & Rasool Shah & Dumitru Baleanu, 2020. "An Efficient Analytical Approach for the Solution of Certain Fractional-Order Dynamical Systems," Energies, MDPI, vol. 13(11), pages 1-14, May.
    2. Wen, Zhenshu, 2020. "The generalized bifurcation method for deriving exact solutions of nonlinear space-time fractional partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 366(C).
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    Cited by:

    1. Remus-Daniel Ene & Nicolina Pop & Marioara Lapadat & Luisa Dungan, 2022. "Approximate Closed-Form Solutions for the Maxwell-Bloch Equations via the Optimal Homotopy Asymptotic Method," Mathematics, MDPI, vol. 10(21), pages 1-13, November.
    2. Yang Zhang & Qiang Gui & Yuzheng Yang & Wei Li, 2022. "The Instability and Response Studies of a Top-Tensioned Riser under Parametric Excitations Using the Differential Quadrature Method," Mathematics, MDPI, vol. 10(8), pages 1-23, April.
    3. Remus-Daniel Ene & Nicolina Pop, 2023. "Semi-Analytical Closed-Form Solutions for the Rikitake-Type System through the Optimal Homotopy Perturbation Method," Mathematics, MDPI, vol. 11(14), pages 1-22, July.

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