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

Farzaneh Safari, Qingshan Tong (), Zhen Tang and Jun Lu
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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
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
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

Mathematics, 2022, vol. 10, issue 21, 1-18

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.

Keywords: weighted Grünwald difference; irregular domain; nonlinear functional diffusion problem; quasilinearization technique (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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