Existence, uniqueness, and galerkin shifted Legendre's approximation of time delays integrodifferential models by adapting the Hilfer fractional attitude

Guaranteeing the uniqueness of the solution will simplify the analysis and provide a clear approximation of the considered problem. This article presents theoretical proof of the presence of a unique solution and leverages approximation for the time delay functions in integrodifferential models in the sense of the Hilfer fractional approach. Once the wellposedness discussion is done, our focus lies on utilizing the Galerkin pseudo-codes based on the OSLPs to generate an approximation by applying GSLM as follows: utilizing the OSLPs to replace the required functions in main Hilfer model, applying the Galerkin pseudo-codes, and transforming Hilfer model into an algebraic system of equations. Herein, the wellposedness is proven by utilizing the fixed point of Banach and the principle of contraction mapping. The proposed scheme is shown to provide efficient, flawless, and expedient solutions for the needed Hilfer fractional delay model. Whilst, the effectiveness of the approach is demonstrated by various delayed examples in comparison with the classical solution. Tabulated and plotted results of the study provide a novel reliable scheme for handling such a Hilfer delayed integrodifferential model. Eventually, valuable noteworthy points and suggestions for future endeavors have been offered.