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A Meshfree Approach for Solving Fractional Galilei Invariant Advection–Diffusion Equation through Weighted–Shifted Grünwald Operator

Author

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  • Farzaneh Safari

    (School of Mathematics and Statistics, Changsha University of Science and Technology, No. 960, 2nd Section, South Wanjiali Road, Tianxin District, Changsha 410004, China
    These authors contributed equally to this work.)

  • Qingshan Tong

    (School of Mathematics and Statistics, Changsha University of Science and Technology, No. 960, 2nd Section, South Wanjiali Road, Tianxin District, Changsha 410004, China
    These authors contributed equally to this work.)

  • Zhen Tang

    (School of Mathematics and Statistics, Changsha University of Science and Technology, No. 960, 2nd Section, South Wanjiali Road, Tianxin District, Changsha 410004, China)

  • Jun Lu

    (Nanjing Hydraulic Research Institute, Hujuguan 34 Road, Nanjing 210024, China)

Abstract

Fractional Galilei invariant advection–diffusion (GIADE) equation, along with its more general version that is the GIADE equation with nonlinear source term, is discretized by coupling weighted and shifted Grünwald difference approximation formulae and Crank–Nicolson technique. The new version of the backward substitution method, a well-established class of meshfree methods, is proposed for a numerical approximation of the consequent equation. In the present approach, the final approximation is given by the summation of the radial basis functions, the primary approximation, and the related correcting functions. Then, the approximation is substituted back to the governing equations where the unknown parameters can be determined. The polynomials, trigonometric functions, multiquadric, or the Gaussian radial basis functions are used in the approximation of the GIADE. Moreover, a quasilinearization technique is employed to transform a nonlinear source term into a linear source term. Finally, three numerical experiments in one and two dimensions are presented to support the method.

Suggested Citation

  • Farzaneh Safari & Qingshan Tong & Zhen Tang & Jun Lu, 2022. "A Meshfree Approach for Solving Fractional Galilei Invariant Advection–Diffusion Equation through Weighted–Shifted Grünwald Operator," Mathematics, MDPI, vol. 10(21), pages 1-18, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4008-:d:956593
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    References listed on IDEAS

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    1. Lin, Ji & Reutskiy, S.Y. & Lu, Jun, 2018. "A novel meshless method for fully nonlinear advection–diffusion-reaction problems to model transfer in anisotropic media," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 459-476.
    2. Lin, Ji & Reutskiy, Sergiy, 2020. "A cubic B-spline semi-analytical algorithm for simulation of 3D steady-state convection-diffusion-reaction problems," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    3. Ndivhuwo Ndou & Phumlani Dlamini & Byron Alexander Jacobs, 2022. "Enhanced Unconditionally Positive Finite Difference Method for Advection–Diffusion–Reaction Equations," Mathematics, MDPI, vol. 10(15), pages 1-18, July.
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