A Matrix Approach by Convolved Fermat Polynomials for Solving the Fractional Burgers’ Equation

Waleed Mohamed Abd-Elhameed (), Omar Mazen Alqubori, Naher Mohammed A. Alsafri, Amr Kamel Amin and Ahmed Gamal Atta
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Waleed Mohamed Abd-Elhameed: Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
Omar Mazen Alqubori: Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
Naher Mohammed A. Alsafri: Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
Amr Kamel Amin: Department of Mathematics, Adham University College, Umm Al-Qura University, Makkah 28653, Saudi Arabia
Ahmed Gamal Atta: Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt

Mathematics, 2025, vol. 13, issue 7, 1-25

Abstract: 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.

Keywords: Fermat polynomials; convolved polynomials; fractional Burgers’ equation; spectral methods; operational matrices; convergence analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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