Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems
The aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equ...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/9/668 |
| _version_ | 1797580041906814976 |
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| author | Abdelfattah Mustafa Ola Ragb Mohamed Salah Reda S. Salama Mokhtar Mohamed |
| author_facet | Abdelfattah Mustafa Ola Ragb Mohamed Salah Reda S. Salama Mokhtar Mohamed |
| author_sort | Abdelfattah Mustafa |
| collection | DOAJ |
| description | The aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equation for convection and diffusion depends on both spatial and transient factors. By using the block marching technique, we transform these equations into an algebraic system using differential quadrature methods and the Caputo-type fractional operator. Next, we develop a MATLAB program that generates code capable of solving the fractional convection–diffusion equation in (1+2) dimensions for each shape function. Our goal is to ensure that our methods are reliable, accurate, efficient, and capable of convergence. To achieve this, we conduct two experiments, comparing the numerical and graphical results with both analytical and numerical solutions. Additionally, we evaluate the accuracy of our findings using the <i>L</i><sub>∞</sub> error. Our tests show that the differential quadrature method, which relies mainly on the discrete singular convolution shape function, is a highly effective numerical approach for fractional convective diffusion problems. It offers superior accuracy, faster convergence, and greater reliability than other techniques. Furthermore, we study the impact of fractional order derivatives, velocity, and positive diffusion parameters on the results. |
| first_indexed | 2024-03-10T22:44:42Z |
| format | Article |
| id | doaj.art-bc6a130df1eb40dc8996addb2c1caa4d |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T22:44:42Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-bc6a130df1eb40dc8996addb2c1caa4d2023-11-19T10:48:32ZengMDPI AGFractal and Fractional2504-31102023-09-017966810.3390/fractalfract7090668Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion ProblemsAbdelfattah Mustafa0Ola Ragb1Mohamed Salah2Reda S. Salama3Mokhtar Mohamed4Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi ArabiaDepartment of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig 44519, EgyptDepartment of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig 44519, EgyptBasic Science Department, Faculty of Engineering, Delta University for Science and Technology, Gamasa 11152, EgyptBasic Science Department, Faculty of Engineering, Delta University for Science and Technology, Gamasa 11152, EgyptThe aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equation for convection and diffusion depends on both spatial and transient factors. By using the block marching technique, we transform these equations into an algebraic system using differential quadrature methods and the Caputo-type fractional operator. Next, we develop a MATLAB program that generates code capable of solving the fractional convection–diffusion equation in (1+2) dimensions for each shape function. Our goal is to ensure that our methods are reliable, accurate, efficient, and capable of convergence. To achieve this, we conduct two experiments, comparing the numerical and graphical results with both analytical and numerical solutions. Additionally, we evaluate the accuracy of our findings using the <i>L</i><sub>∞</sub> error. Our tests show that the differential quadrature method, which relies mainly on the discrete singular convolution shape function, is a highly effective numerical approach for fractional convective diffusion problems. It offers superior accuracy, faster convergence, and greater reliability than other techniques. Furthermore, we study the impact of fractional order derivatives, velocity, and positive diffusion parameters on the results.https://www.mdpi.com/2504-3110/7/9/668fractional derivativeblock marching methoddifferential quadrature techniquediscrete singular convolutionCaputofractional convective diffusion |
| spellingShingle | Abdelfattah Mustafa Ola Ragb Mohamed Salah Reda S. Salama Mokhtar Mohamed Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems Fractal and Fractional fractional derivative block marching method differential quadrature technique discrete singular convolution Caputo fractional convective diffusion |
| title | Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems |
| title_full | Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems |
| title_fullStr | Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems |
| title_full_unstemmed | Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems |
| title_short | Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems |
| title_sort | distinctive shape functions of fractional differential quadrature for solving two dimensional space fractional diffusion problems |
| topic | fractional derivative block marching method differential quadrature technique discrete singular convolution Caputo fractional convective diffusion |
| url | https://www.mdpi.com/2504-3110/7/9/668 |
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