A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential Equations
Background: In many studies, spectral methods based on one of the orthogonal polynomials and weighted residual methods (WRMs) have been used to convert the variable-order fractional differential equation (VO-FDEs) into a system of linear or nonlinear algebraic equations, and then solve this system...
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Mustansiriyah University
2024-12-01
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| Series: | Al-Mustansiriyah Journal of Science |
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| Online Access: | https://mjs.uomustansiriyah.edu.iq/index.php/MJS/article/view/1564 |
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| author | Abdulrazzaq T. Abed Ekhlass S. Al-Rawi |
| author_facet | Abdulrazzaq T. Abed Ekhlass S. Al-Rawi |
| author_sort | Abdulrazzaq T. Abed |
| collection | DOAJ |
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Background: In many studies, spectral methods based on one of the orthogonal polynomials and weighted residual methods (WRMs) have been used to convert the variable-order fractional differential equation (VO-FDEs) into a system of linear or nonlinear algebraic equations, and then solve this system to obtain the approximate solution. Objective: In this paper, a numerical method is presented for solving VO-FDEs. The proposed method is based on Chelyshkov polynomials (CPs). Methods: WRMs are used to obtain approximate solutions of the governing differential equations. In addition, a new weight function based on Mittag-Leffler functions is proposed. The proposed method is applied to a group of linear and non-linear examples with initial and boundary conditions. Results: Acceptable results are obtained in most of the examples. In addition, the effect of polynomials (such as Chebyshev, Jacobi, Legendre, Gegenbauer, Hermite, Taylor, Mittag-Leffler, and Bernstein polynomials) on the accuracy of the approximate solution is studied, and their effect was found to be minimal in most tests. The effect of the proposed weight function is also studied in comparison with the weight functions presented in WRMs, and it was found that it has a strong effect in most examples. Conclusions: The results obtained indicate that the proposed method is effective for solving equations of variable order.
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| format | Article |
| id | doaj-art-e2d6dd7bc19147568792e8bfb5f450bb |
| institution | Kabale University |
| issn | 1814-635X 2521-3520 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Mustansiriyah University |
| record_format | Article |
| series | Al-Mustansiriyah Journal of Science |
| spelling | doaj-art-e2d6dd7bc19147568792e8bfb5f450bb2025-01-03T05:01:21ZengMustansiriyah UniversityAl-Mustansiriyah Journal of Science1814-635X2521-35202024-12-0135410.23851/mjs.v35i4.1564A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential EquationsAbdulrazzaq T. Abed0https://orcid.org/0000-0003-0521-5476Ekhlass S. Al-Rawi 1https://orcid.org/0000-0001-6940-7050Department of Mathematics, College of Education for Pure Science, University of Mosul, Mosul, IraqDepartment of Mathematics, College of Computer Sciences and Mathematics, University of Mosul, Mosul, Iraq Background: In many studies, spectral methods based on one of the orthogonal polynomials and weighted residual methods (WRMs) have been used to convert the variable-order fractional differential equation (VO-FDEs) into a system of linear or nonlinear algebraic equations, and then solve this system to obtain the approximate solution. Objective: In this paper, a numerical method is presented for solving VO-FDEs. The proposed method is based on Chelyshkov polynomials (CPs). Methods: WRMs are used to obtain approximate solutions of the governing differential equations. In addition, a new weight function based on Mittag-Leffler functions is proposed. The proposed method is applied to a group of linear and non-linear examples with initial and boundary conditions. Results: Acceptable results are obtained in most of the examples. In addition, the effect of polynomials (such as Chebyshev, Jacobi, Legendre, Gegenbauer, Hermite, Taylor, Mittag-Leffler, and Bernstein polynomials) on the accuracy of the approximate solution is studied, and their effect was found to be minimal in most tests. The effect of the proposed weight function is also studied in comparison with the weight functions presented in WRMs, and it was found that it has a strong effect in most examples. Conclusions: The results obtained indicate that the proposed method is effective for solving equations of variable order. https://mjs.uomustansiriyah.edu.iq/index.php/MJS/article/view/1564Variable order derivativeWeighted residual method Spectral methodMittag-Leffler weight function Chelyshkov polynomials |
| spellingShingle | Abdulrazzaq T. Abed Ekhlass S. Al-Rawi A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential Equations Al-Mustansiriyah Journal of Science Variable order derivative Weighted residual method Spectral method Mittag-Leffler weight function Chelyshkov polynomials |
| title | A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential Equations |
| title_full | A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential Equations |
| title_fullStr | A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential Equations |
| title_full_unstemmed | A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential Equations |
| title_short | A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential Equations |
| title_sort | new spectral mittag leffler weighted method to solve variable order fractional differential equations |
| topic | Variable order derivative Weighted residual method Spectral method Mittag-Leffler weight function Chelyshkov polynomials |
| url | https://mjs.uomustansiriyah.edu.iq/index.php/MJS/article/view/1564 |
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