Delay volterra integrodifferential models of fractional orders and exponential kernels: Well-posedness theoretical results and Legendre–Galerkin shifted approximations

The utilized research work studies the well-posedness theoretical results and Galerkin-shifted approximations of delay Volterra integrodifferential models of fractional orders and exponential kernels in linear and nonlinear types in the Caputo–Fabrizio sense. Schauder’s and Arzela–Ascoli’s theorems are employed to prove a local existence theorem. After that, the Laplace continuous transform is employed to establish and confirm the global well-posedness theoretical results for the presented delay model. For numerical solutions, the Legendre–Galerkin shifted approximations’ algorithm has been used by the dependence on the orthogonal Legendre shifted polynomials to find out the required approximations. Utilizing this scheme, the considered delay model is reduced to an algebraic system with initial constraints. The algorithm precision, error behavior, and convergence are studied. Several numerical experiments together with several tables and figures are included to present the performance and validation of the algorithm presented. At last, some consequences and notes according to the given consequences were offered in the final section with several suggestions to guide future action.