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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Waleed Mohamed Abd-Elhameed & Omar Mazen Alqubori & Naher Mohammed A. Alsafri & Amr Kamel Amin & Ahmed Gamal Atta, 2025. "A Matrix Approach by Convolved Fermat Polynomials for Solving the Fractional Burgers’ Equation," Mathematics, MDPI, vol. 13(7), pages 1-25, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1135-:d:1624063
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    References listed on IDEAS

    as
    1. Postavaru, Octavian, 2023. "An efficient numerical method based on Fibonacci polynomials to solve fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 406-422.
    2. Waleed Mohamed Abd-Elhameed & Omar Mazen Alqubori & Anna Napoli, 2024. "On Convolved Fibonacci Polynomials," Mathematics, MDPI, vol. 13(1), pages 1-25, December.
    3. Waleed Mohamed Abd-Elhameed & Omar Mazen Alqubori & Ahmed Gamal Atta, 2024. "A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials," Mathematics, MDPI, vol. 12(23), pages 1-18, November.
    4. Waleed Mohamed Abd-Elhameed & Omar Mazen Alqubori & Ahmed Gamal Atta, 2025. "A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials," Mathematics, MDPI, vol. 13(2), pages 1-26, January.
    5. Şahin, Adem & Ramírez, José L., 2016. "Determinantal and permanental representations of convolved Lucas polynomials," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 314-322.
    6. Dwivedi, Himanshu Kumar & Rajeev,, 2025. "A novel fast second order approach with high-order compact difference scheme and its analysis for the tempered fractional Burgers equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 227(C), pages 168-188.
    7. Hossein Jafari & Hassan Kamil Jassim & Ali Ansari & Van Thinh Nguyen, 2024. "Local Fractional Variational Iteration Transform Method: A Tool For Solving Local Fractional Partial Differential Equations," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 32(04), pages 1-8.
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