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A high order numerical method and its convergence for time-fractional fourth order partial differential equations

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  • Roul, Pradip
  • Prasad Goura, V.M.K.

Abstract

This paper is concerned with design and analysis of a high order numerical approach based on a uniform mesh to approximate the solution of time-fractional fourth order partial differential equations. In this approach, we first approximate the time-fractional derivative appearing in the governing equation by means of Caputo’s definition and then construct a sextic B-spline collocation method for solving the resulting equation. It is proved that the present method is unconditionally stable. Convergence analysis of the method is discussed. We prove that the method is (2−α)th order convergence with respect to time variable and fourth order convergence with respect to space variable. Three test problems are considered to demonstrate the applicability and efficiency of the new method. It is shown that the rate of convergence predicted theoretically is the same as that obtained experimentally. Numerical results have been compared with those reported previously in literature. It is shown that our method yields more accurate results when compared to the existing methods.

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  • Roul, Pradip & Prasad Goura, V.M.K., 2020. "A high order numerical method and its convergence for time-fractional fourth order partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 366(C).
    • Handle: RePEc:eee:apmaco:v:366:y:2020:i:c:s0096300319307192
    DOI: 10.1016/j.amc.2019.124727
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    References listed on IDEAS

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    1. Roul, Pradip & Madduri, Harshita & Kassner, Klaus, 2019. "A sixth-order numerical method for a strongly nonlinear singular boundary value problem governing electrohydrodynamic flow in a circular cylindrical conduit," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 416-433.
    2. Al-Smadi, Mohammed & Arqub, Omar Abu, 2019. "Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 280-294.
    3. Roul, Pradip & Prasad Goura, V.M.K., 2019. "B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 428-450.
    4. Goura, V.M.K. Prasad & Roul, Pradip, 2019. "Erratum to: B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 198-201.
    5. S. C. S. Rao & M. Kumar, 2007. "B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems," Journal of Optimization Theory and Applications, Springer, vol. 134(1), pages 91-105, July.
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    Cited by:

    1. Liu, Li & Fan, Zhenbin & Li, Gang & Piskarev, Sergey, 2021. "Discrete almost maximal regularity and stability for fractional differential equations in Lp([0, 1], Ω)," Applied Mathematics and Computation, Elsevier, vol. 389(C).

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