Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs

Zahrah I. Salman, Majid Tavassoli Kajani (), Mohammed Sahib Mechee and Masoud Allame
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Zahrah I. Salman: Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan P.O. Box 158-81595, Iran
Majid Tavassoli Kajani: Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan P.O. Box 158-81595, Iran
Mohammed Sahib Mechee: Information Technology Research and Development Center, University of Kufa, Najaf 540011, Iraq
Masoud Allame: Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan P.O. Box 158-81595, Iran

Mathematics, 2023, vol. 11, issue 17, 1-15

Abstract: Proposing a matrix transform method to solve a fractional partial differential equation is the main aim of this paper. The main model can be transferred to a partial-integro differential equation (PIDE) with a weakly singular kernel. The spatial direction is approximated by a fourth-order difference scheme. Also, the temporal derivative is discretized via a second-order numerical procedure. First, the spatial derivatives are approximated by a fourth-order operator to compute the second-order derivatives. This process produces a system of differential equations related to the time variable. Then, the Crank–Nicolson idea is utilized to achieve a full-discrete scheme. The kernel of the integral term is discretized by using the Lagrange polynomials to overcome its singularity. Subsequently, we prove the convergence and stability of the new difference scheme by utilizing the Rayleigh–Ritz theorem. Finally, some numerical examples in one-dimensional and two-dimensional cases are presented to verify the theoretical results.

Keywords: matrix transform method; fourth-order difference scheme; partial-integro differential equation; Rayleigh–Ritz theorem; error estimate (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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