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Volume 12
Issue 23
10.3390/math12233672
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

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

by
Waleed Mohamed Abd-Elhameed
1,*,
Omar Mazen Alqubori
1 and
Ahmed Gamal Atta
2
1
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
2
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(23), 3672; https://doi.org/10.3390/math12233672 (registering DOI)
Submission received: 5 September 2024 / Revised: 10 November 2024 / Accepted: 22 November 2024 / Published: 23 November 2024
(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)
Versions Notes

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.
Keywords: Lucas polynomials; shifted polynomials; derivative formulas; operational matrices; numerical solutions Lucas polynomials; shifted polynomials; derivative formulas; operational matrices; numerical solutions

Share and Cite

MDPI and ACS Style

Abd-Elhameed, W.M.; Alqubori, O.M.; Atta, A.G. A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials. Mathematics 2024, 12, 3672. https://doi.org/10.3390/math12233672

AMA Style

Abd-Elhameed WM, Alqubori OM, Atta AG. A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials. Mathematics. 2024; 12(23):3672. https://doi.org/10.3390/math12233672

Chicago/Turabian Style

Abd-Elhameed, Waleed Mohamed, Omar Mazen Alqubori, and Ahmed Gamal Atta. 2024. "A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials" Mathematics 12, no. 23: 3672. https://doi.org/10.3390/math12233672

APA Style

Abd-Elhameed, W. M., Alqubori, O. M., & Atta, A. G. (2024). A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials. Mathematics, 12(23), 3672. https://doi.org/10.3390/math12233672

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