Analysis And Collocation Approximations Using Shifted Legendre Galerkin Algorithm For Systems Of Fractional Integrodifferential Delay Problems Regarding Hilfer Fractional Type

Delay problems are a special type of equation that includes values of the dependent variable at earlier times; it is widely used in modeling many natural and engineering phenomena that involve delays. This pioneering analysis initially addressed the existence and uniqueness attitude (EUA) of solutions for systems of fractional integro differential delay problems (FIDDPs) using the Hilfer fractional type. At first, we explored the properties of fundamental definitions concerning fractional attitudes, leveraging their characteristics to convert the system of FIDDP into an equivalent system of fractional volterra differential problem (FVDS). Employing the axiom of contraction mapping, we established the EUA of a solution for the resulting delays set. Thereafter, for approximation aims, the Galerkin scheme with shifted legendre orthogonal polynomials (SLOPs) was implemented. This approach transforms the system of FIDDP into a series of algebraic equations, enabling the approximation of the prompted solutions. A significant advantage of this algorithm lies in its ability to get accurate results by using fewer iterations compared to the difficulty of delays. To authenticate the workability of the theoretical–numerical achievements, we performed extensive tests on various delay problems of numerous natures. The attained achievements, presented with figures and tables, showcase and conduct the algorithm’s superior accuracy. The conclusion summarizes diverse highlights and proposes potential areas for future research.