A Fourth-Order Time-Stepping Method for Two-Dimensional, Distributed-Order, Space-Fractional, Inhomogeneous Parabolic Equations
Distributed-order, space-fractional diffusion equations are used to describe physical processes that lack power-law scaling. A fourth-order-accurate, <i>A</i>-stable time-stepping method was developed, analyzed, and implemented to solve inhomogeneous parabolic problems having Riesz-space...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-10-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/10/592 |
| _version_ | 1797473252713431040 |
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| author | Muhammad Yousuf Khaled M. Furati Abdul Q. M. Khaliq |
| author_facet | Muhammad Yousuf Khaled M. Furati Abdul Q. M. Khaliq |
| author_sort | Muhammad Yousuf |
| collection | DOAJ |
| description | Distributed-order, space-fractional diffusion equations are used to describe physical processes that lack power-law scaling. A fourth-order-accurate, <i>A</i>-stable time-stepping method was developed, analyzed, and implemented to solve inhomogeneous parabolic problems having Riesz-space-fractional, distributed-order derivatives. The considered problem was transformed into a multi-term, space-fractional problem using Simpson’s three-eighths rule. The method is based on an approximation of matrix exponential functions using fourth-order diagonal Padé approximation. The Gaussian quadrature approach is used to approximate the integral matrix exponential function, along with the inhomogeneous term. Partial fraction splitting is used to address the issues regarding stability and computational efficiency. Convergence of the method was proved analytically and demonstrated through numerical experiments. CPU time was recorded in these experiments to show the computational efficiency of the method. |
| first_indexed | 2024-03-09T20:12:07Z |
| format | Article |
| id | doaj.art-db874ca08333450d9deec8fbd516d3b7 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T20:12:07Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-db874ca08333450d9deec8fbd516d3b72023-11-24T00:12:11ZengMDPI AGFractal and Fractional2504-31102022-10-0161059210.3390/fractalfract6100592A Fourth-Order Time-Stepping Method for Two-Dimensional, Distributed-Order, Space-Fractional, Inhomogeneous Parabolic EquationsMuhammad Yousuf0Khaled M. Furati1Abdul Q. M. Khaliq2Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi ArabiaDepartment of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi ArabiaDepartment of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 37132-0001, USADistributed-order, space-fractional diffusion equations are used to describe physical processes that lack power-law scaling. A fourth-order-accurate, <i>A</i>-stable time-stepping method was developed, analyzed, and implemented to solve inhomogeneous parabolic problems having Riesz-space-fractional, distributed-order derivatives. The considered problem was transformed into a multi-term, space-fractional problem using Simpson’s three-eighths rule. The method is based on an approximation of matrix exponential functions using fourth-order diagonal Padé approximation. The Gaussian quadrature approach is used to approximate the integral matrix exponential function, along with the inhomogeneous term. Partial fraction splitting is used to address the issues regarding stability and computational efficiency. Convergence of the method was proved analytically and demonstrated through numerical experiments. CPU time was recorded in these experiments to show the computational efficiency of the method.https://www.mdpi.com/2504-3110/6/10/592distributed-orderRiesz-space-fractional diffusionPadé approximationsplitting technique |
| spellingShingle | Muhammad Yousuf Khaled M. Furati Abdul Q. M. Khaliq A Fourth-Order Time-Stepping Method for Two-Dimensional, Distributed-Order, Space-Fractional, Inhomogeneous Parabolic Equations Fractal and Fractional distributed-order Riesz-space-fractional diffusion Padé approximation splitting technique |
| title | A Fourth-Order Time-Stepping Method for Two-Dimensional, Distributed-Order, Space-Fractional, Inhomogeneous Parabolic Equations |
| title_full | A Fourth-Order Time-Stepping Method for Two-Dimensional, Distributed-Order, Space-Fractional, Inhomogeneous Parabolic Equations |
| title_fullStr | A Fourth-Order Time-Stepping Method for Two-Dimensional, Distributed-Order, Space-Fractional, Inhomogeneous Parabolic Equations |
| title_full_unstemmed | A Fourth-Order Time-Stepping Method for Two-Dimensional, Distributed-Order, Space-Fractional, Inhomogeneous Parabolic Equations |
| title_short | A Fourth-Order Time-Stepping Method for Two-Dimensional, Distributed-Order, Space-Fractional, Inhomogeneous Parabolic Equations |
| title_sort | fourth order time stepping method for two dimensional distributed order space fractional inhomogeneous parabolic equations |
| topic | distributed-order Riesz-space-fractional diffusion Padé approximation splitting technique |
| url | https://www.mdpi.com/2504-3110/6/10/592 |
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