A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems-Reference-Cited by-同舟云学术

A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems

Published:2023-08-25 Issue:9 Volume:7 Page:649
ISSN:2504-3110
Container-title:Fractal and Fractional
language:en
Short-container-title:Fractal Fract

Author:

Derakhshan Mohammadhossein¹²^ORCID,Hendy Ahmed S.³⁴^ORCID,Lopes António M.⁵^ORCID,Galhano Alexandra⁶^ORCID,Zaky Mahmoud A.⁷^ORCID

Affiliation:

1. Department of Industrial Engineering, Apadana Institute of Higher Education, Shiraz 7187985443, Iran

2. Faculty of Technology and Engineering, Zand Institute of Higher Education, Shiraz 8415683111, Iran

3. Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia

4. Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt

5. LAETA/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

6. Faculdade de Ciências Naturais, Engenharias e Tecnologias, Universidade Lusófona do Porto, Rua de Augusto Rosa 24, 4000-098 Porto, Portugal

7. Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia

Abstract

Invoking the matrix transfer technique, we propose a novel numerical scheme to solve the time-fractional advection–dispersion equation (ADE) with distributed-order Riesz-space fractional derivatives (FDs). The method adopts the midpoint rule to reformulate the distributed-order Riesz-space FDs by means of a second-order linear combination of Riesz-space FDs. Then, a central difference approximation is used side by side with the matrix transform technique for approximating the Riesz-space FDs. Based on this, the distributed-order time-fractional ADE is transformed into a time-fractional ordinary differential equation in the Caputo sense, which has an equivalent Volterra integral form. The Simpson method is used to discretize the weakly singular kernel of the resulting Volterra integral equation. Stability, convergence, and error analysis are presented. Finally, simulations are performed to substantiate the theoretical findings.

Funder

Imam Mohammad Ibn Saud Islamic University

Publisher

MDPI AG

Subject

Statistics and Probability,Statistical and Nonlinear Physics,Analysis

Link

https://www.mdpi.com/2504-3110/7/9/649/pdf

Reference32 articles.

1. Podulbny, I. (1999). Fractional Differential Equations, Academic Press.

2. Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier.

3. Some aspects of fractional diffusion equations of single and distributed order;Mainardi;Appl. Math. Comput.,2007

4. Derakhshan, M., and Aminataei, A. (2021). New approach for the chaotic dynamical systems involving Caputo-Prabhakar fractional derivative using Adams-Bashforth scheme. J. Differ. Equ. Appl., 1–17.

5. New generalized Mellin transform and applications to partial and fractional differential equations;Ata;Int. J. Math. Comput. Eng.,2023