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

This paper considers linear and nonlinear fractional delay Volterra integrodiffierential equation of order rho 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.