Fractional delay integrodifferential equations of nonsingular kernels: Existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials

Hind Sweis, Omar Abu Arqub and Nabil Shawagfeh
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Hind Sweis: Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan
Omar Abu Arqub: ��Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan
Nabil Shawagfeh: Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan

International Journal of Modern Physics C (IJMPC), 2023, vol. 34, issue 04, 1-23

Abstract: This paper considers linear and nonlinear fractional delay Volterra integrodifferential equation of order Ï in the Atangana–Beleanu–Caputo (ABC) sense. We used continuous Laplace transform (CLT) to find equivalent Volterra integral equations that have been used together with the Arzela–Ascoli theorem and Schauder’s fixed point theorem to prove the local existence solution. Moreover, the obtained Volterra integral equations and the contraction mapping theorem have been successfully applied to construct and prove the global existence and uniqueness of the solution for the considered fractional delay integrodifferential equation (FDIDE). The Galerkin algorithm instituted within shifted Legendre polynomials (SLPs) is applied in the approximation procedure for the corresponding delay equation. Indeed, by this algorithm, we get algebraic system models and by solving this system we gained the approximated nodal solution. The reliability of the method and reduction in the size of the computational work give the algorithm wider applicability. Linear and nonlinear examples are included with some tables and figures to show the effectiveness of the method in comparison with the exact solutions. Finally, some valuable notes and details extracted from the presented results were presented in the last part, with the sign to some of our future works.

Keywords: Fractional delay integrodifferential equation; Atangana–Baleanu–Caputo derivative; Banach fixed point theorem; contraction mapping theorem; Galerkin method; shifted Legendre polynomial (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0129183123500523

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