Well-posedness analysis and pseudo-Galerkin approximations using Tau Legendre algorithm for fractional systems of delay differential models regarding Hilfer (α,β)-framework set.

Fractional calculus serves as a versatile and potent tool for the modeling and control of intricate systems. This discussion debates the system of DFDEs with two regimes; theoretically and numerically. For theoretical analysis, we have established the EUE by leveraging the definition of Hilfer (α,β)-framework.

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.