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A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials

Author

Listed:
  • Waleed Mohamed Abd-Elhameed

    (Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia)

  • Omar Mazen Alqubori

    (Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia)

  • Ahmed Gamal Atta

    (Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt)

Abstract

This work employs newly shifted Lucas polynomials to approximate solutions to the time-fractional Fitzhugh–Nagumo differential equation (TFFNDE) relevant to neuroscience. Novel essential formulae for the shifted Lucas polynomials are crucial for developing our suggested numerical approach. The analytic and inversion formulas are introduced, and after that, new formulas that express these polynomials’ integer and fractional derivatives are derived to facilitate the construction of integer and fractional operational matrices for the derivatives. Employing these operational matrices with the typical collocation method converts the TFFNDE into a system of algebraic equations that can be addressed with standard numerical solvers. The convergence analysis of the shifted Lucas expansion is carefully investigated. Certain inequalities involving the golden ratio are established in this context. The suggested numerical method is evaluated using several numerical examples to verify its applicability and efficiency.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3672-:d:1527836
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    References listed on IDEAS

    as
    1. Abdelkawy, M.A. & Lopes, António M. & Babatin, Mohammed M., 2020. "Shifted fractional Jacobi collocation method for solving fractional functional differential equations of variable order," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    2. Qiu, Wenlin & Chen, Hongbin & Zheng, Xuan, 2019. "An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 298-314.
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