A Matrix Approach by Convolved Fermat Polynomials for Solving the Fractional Burgers’ Equation
This article employs certain polynomials that generalize standard Fermat polynomials, called convolved Fermat polynomials, to numerically solve the fractional Burgers’ equation. New theoretical results of these polynomials are developed and utilized along with the collocation method to find approxim...
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MDPI AG
2025-03-01
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| author | Waleed Mohamed Abd-Elhameed Omar Mazen Alqubori Naher Mohammed A. Alsafri Amr Kamel Amin Ahmed Gamal Atta |
| author_facet | Waleed Mohamed Abd-Elhameed Omar Mazen Alqubori Naher Mohammed A. Alsafri Amr Kamel Amin Ahmed Gamal Atta |
| author_sort | Waleed Mohamed Abd-Elhameed |
| collection | DOAJ |
| description | This article employs certain polynomials that generalize standard Fermat polynomials, called convolved Fermat polynomials, to numerically solve the fractional Burgers’ equation. New theoretical results of these polynomials are developed and utilized along with the collocation method to find approximate solutions of the fractional Burgers’ equation. The basic idea behind the proposed numerical algorithm is based on establishing the operational matrices of derivatives of both integer and fractional derivatives of the convolved Fermat polynomials that help to convert the equation governed by its underlying conditions into an algebraic system of equations that can be treated numerically. A comprehensive study is performed to analyze the error of the proposed convolved Fermat expansion. Some numerical examples are presented to test our proposed numerical algorithm, and some comparisons are made. The results indicate that the proposed algorithm is applicable and accurate. |
| format | Article |
| id | doaj-art-e840023201304da9a6b2f487ebb7328e |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-e840023201304da9a6b2f487ebb7328e2025-08-20T03:08:53ZengMDPI AGMathematics2227-73902025-03-01137113510.3390/math13071135A Matrix Approach by Convolved Fermat Polynomials for Solving the Fractional Burgers’ EquationWaleed Mohamed Abd-Elhameed0Omar Mazen Alqubori1Naher Mohammed A. Alsafri2Amr Kamel Amin3Ahmed Gamal Atta4Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptDepartment of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi ArabiaDepartment of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi ArabiaDepartment of Mathematics, Adham University College, Umm Al-Qura University, Makkah 28653, Saudi ArabiaDepartment of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, EgyptThis article employs certain polynomials that generalize standard Fermat polynomials, called convolved Fermat polynomials, to numerically solve the fractional Burgers’ equation. New theoretical results of these polynomials are developed and utilized along with the collocation method to find approximate solutions of the fractional Burgers’ equation. The basic idea behind the proposed numerical algorithm is based on establishing the operational matrices of derivatives of both integer and fractional derivatives of the convolved Fermat polynomials that help to convert the equation governed by its underlying conditions into an algebraic system of equations that can be treated numerically. A comprehensive study is performed to analyze the error of the proposed convolved Fermat expansion. Some numerical examples are presented to test our proposed numerical algorithm, and some comparisons are made. The results indicate that the proposed algorithm is applicable and accurate.https://www.mdpi.com/2227-7390/13/7/1135Fermat polynomialsconvolved polynomialsfractional Burgers’ equationspectral methodsoperational matricesconvergence analysis |
| spellingShingle | Waleed Mohamed Abd-Elhameed Omar Mazen Alqubori Naher Mohammed A. Alsafri Amr Kamel Amin Ahmed Gamal Atta A Matrix Approach by Convolved Fermat Polynomials for Solving the Fractional Burgers’ Equation Mathematics Fermat polynomials convolved polynomials fractional Burgers’ equation spectral methods operational matrices convergence analysis |
| title | A Matrix Approach by Convolved Fermat Polynomials for Solving the Fractional Burgers’ Equation |
| title_full | A Matrix Approach by Convolved Fermat Polynomials for Solving the Fractional Burgers’ Equation |
| title_fullStr | A Matrix Approach by Convolved Fermat Polynomials for Solving the Fractional Burgers’ Equation |
| title_full_unstemmed | A Matrix Approach by Convolved Fermat Polynomials for Solving the Fractional Burgers’ Equation |
| title_short | A Matrix Approach by Convolved Fermat Polynomials for Solving the Fractional Burgers’ Equation |
| title_sort | matrix approach by convolved fermat polynomials for solving the fractional burgers equation |
| topic | Fermat polynomials convolved polynomials fractional Burgers’ equation spectral methods operational matrices convergence analysis |
| url | https://www.mdpi.com/2227-7390/13/7/1135 |
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