Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations

Authors

  • S. M. Sayed Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt
  • A. S. Mohamed Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt
  • E. M. Abo El-Dahab Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt
  • Y. H. Youssri Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt https://orcid.org/0000-0003-0403-8797

DOI:

https://doi.org/10.37256/cm.5220244559

Keywords:

shifted gegenbauer polynomials, boundary value problems, fractional differential equations, bagley-trovik equation, spectral methods, convergence analysis

Abstract

This article introduces two numerical methods to address boundary value problems associated with secondorder and fractional differential equations. These methods employ two parameters related to shifted Gegenbauer polynomials as their basis functions. The process involves establishing a differentiation operational matrix for the shifted Gegenbauer polynomials. Subsequently, the initial/boundary value problems for ordinary and fractional differential equations are transformed into a system of equations through the Galerkin, collocation, and tau methods. The convergence analysis is ensured by leveraging theorems pertaining to the shifted Gegenbauer polynomials. To validate the accuracy of the approach, numerous numerical examples are presented.

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Published

2024-05-10

How to Cite

1.
Sayed SM, Mohamed AS, El-Dahab EMA, Youssri YH. Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations. Contemp. Math. [Internet]. 2024 May 10 [cited 2024 May 28];5(2):4123-49. Available from: https://ojs.wiserpub.com/index.php/CM/article/view/4559