A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems
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 me...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-08-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/9/649 |
| _version_ | 1797580038278742016 |
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| author | Mohammadhossein Derakhshan Ahmed S. Hendy António M. Lopes Alexandra Galhano Mahmoud A. Zaky |
| author_facet | Mohammadhossein Derakhshan Ahmed S. Hendy António M. Lopes Alexandra Galhano Mahmoud A. Zaky |
| author_sort | Mohammadhossein Derakhshan |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T22:44:37Z |
| format | Article |
| id | doaj.art-ab79f1cb715b4318b5e48ca2b4c91a83 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T22:44:37Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-ab79f1cb715b4318b5e48ca2b4c91a832023-11-19T10:48:16ZengMDPI AGFractal and Fractional2504-31102023-08-017964910.3390/fractalfract7090649A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion ProblemsMohammadhossein Derakhshan0Ahmed S. Hendy1António M. Lopes2Alexandra Galhano3Mahmoud A. Zaky4Department of Industrial Engineering, Apadana Institute of Higher Education, Shiraz 7187985443, IranComputational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, RussiaLAETA/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, PortugalFaculdade de Ciências Naturais, Engenharias e Tecnologias, Universidade Lusófona do Porto, Rua de Augusto Rosa 24, 4000-098 Porto, PortugalDepartment of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi ArabiaInvoking 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.https://www.mdpi.com/2504-3110/7/9/649advection–dispersion equationmatrix transform methodconvergence analysisdistributed-orderRiesz fractional derivative |
| spellingShingle | Mohammadhossein Derakhshan Ahmed S. Hendy António M. Lopes Alexandra Galhano Mahmoud A. Zaky A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems Fractal and Fractional advection–dispersion equation matrix transform method convergence analysis distributed-order Riesz fractional derivative |
| title | A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems |
| title_full | A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems |
| title_fullStr | A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems |
| title_full_unstemmed | A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems |
| title_short | A Matrix Transform Technique for Distributed-Order Time-Fractional Advection–Dispersion Problems |
| title_sort | matrix transform technique for distributed order time fractional advection dispersion problems |
| topic | advection–dispersion equation matrix transform method convergence analysis distributed-order Riesz fractional derivative |
| url | https://www.mdpi.com/2504-3110/7/9/649 |
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