Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation
Here, we provide a new method to solve the time-fractional diffusion equation (TFDE) following the spectral tau approach. Our proposed numerical solution is expressed in terms of a double Lucas expansion. The discretization of the technique is based on several formulas about Lucas polynomials, such...
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AIMS Press
2024-12-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241646 |
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| author | Waleed Mohamed Abd-Elhameed Abdullah F. Abu Sunayh Mohammed H. Alharbi Ahmed Gamal Atta |
| author_facet | Waleed Mohamed Abd-Elhameed Abdullah F. Abu Sunayh Mohammed H. Alharbi Ahmed Gamal Atta |
| author_sort | Waleed Mohamed Abd-Elhameed |
| collection | DOAJ |
| description | Here, we provide a new method to solve the time-fractional diffusion equation (TFDE) following the spectral tau approach. Our proposed numerical solution is expressed in terms of a double Lucas expansion. The discretization of the technique is based on several formulas about Lucas polynomials, such as those for explicit integer and fractional derivatives, products, and certain definite integrals of these polynomials. These formulas aid in transforming the TFDE and its conditions into a matrix system that can be treated through a suitable numerical procedure. We conduct a study on the convergence analysis of the double Lucas expansion. In addition, we provide a few examples to ensure that the proposed numerical approach is applicable and efficient. |
| format | Article |
| id | doaj-art-3e05e738ef204116bfbc0825455d1b32 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-3e05e738ef204116bfbc0825455d1b322025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912345673458710.3934/math.20241646Spectral tau technique via Lucas polynomials for the time-fractional diffusion equationWaleed Mohamed Abd-Elhameed0Abdullah F. Abu Sunayh1Mohammed H. Alharbi2Ahmed Gamal Atta3Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptDepartment of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi ArabiaDepartment of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi ArabiaDepartment of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo, EgyptHere, we provide a new method to solve the time-fractional diffusion equation (TFDE) following the spectral tau approach. Our proposed numerical solution is expressed in terms of a double Lucas expansion. The discretization of the technique is based on several formulas about Lucas polynomials, such as those for explicit integer and fractional derivatives, products, and certain definite integrals of these polynomials. These formulas aid in transforming the TFDE and its conditions into a matrix system that can be treated through a suitable numerical procedure. We conduct a study on the convergence analysis of the double Lucas expansion. In addition, we provide a few examples to ensure that the proposed numerical approach is applicable and efficient.https://www.aimspress.com/article/doi/10.3934/math.20241646time-fractional diffusion equationlucas polynomialsspectral methodserror bound |
| spellingShingle | Waleed Mohamed Abd-Elhameed Abdullah F. Abu Sunayh Mohammed H. Alharbi Ahmed Gamal Atta Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation AIMS Mathematics time-fractional diffusion equation lucas polynomials spectral methods error bound |
| title | Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation |
| title_full | Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation |
| title_fullStr | Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation |
| title_full_unstemmed | Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation |
| title_short | Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation |
| title_sort | spectral tau technique via lucas polynomials for the time fractional diffusion equation |
| topic | time-fractional diffusion equation lucas polynomials spectral methods error bound |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241646 |
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