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Volume 11
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10.3390/math11173786
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Article

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

by
Zahrah I. Salman
1,
Majid Tavassoli Kajani
1,*,
Mohammed Sahib Mechee
2 and
Masoud Allame
1
1
Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan P.O. Box 158-81595, Iran
2
Information Technology Research and Development Center, University of Kufa, Najaf 540011, Iraq
*
Author to whom correspondence should be addressed.
Mathematics 2023, 11(17), 3786; https://doi.org/10.3390/math11173786
Submission received: 10 July 2023 / Revised: 17 August 2023 / Accepted: 21 August 2023 / Published: 3 September 2023
(This article belongs to the Section E: Applied Mathematics)
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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 matrix transform method; fourth-order difference scheme; partial-integro differential equation; Rayleigh–Ritz theorem; error estimate

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MDPI and ACS Style

Salman, Z.I.; Tavassoli Kajani, M.; Mechee, M.S.; Allame, M. Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs. Mathematics 2023, 11, 3786. https://doi.org/10.3390/math11173786

AMA Style

Salman ZI, Tavassoli Kajani M, Mechee MS, Allame M. Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs. Mathematics. 2023; 11(17):3786. https://doi.org/10.3390/math11173786

Chicago/Turabian Style

Salman, Zahrah I., Majid Tavassoli Kajani, Mohammed Sahib Mechee, and Masoud Allame. 2023. "Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs" Mathematics 11, no. 17: 3786. https://doi.org/10.3390/math11173786

APA Style

Salman, Z. I., Tavassoli Kajani, M., Mechee, M. S., & Allame, M. (2023). Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs. Mathematics, 11(17), 3786. https://doi.org/10.3390/math11173786

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