Hilfer fractional delay differential equations: Existence and uniqueness computational results and pointwise approximation utilizing the Shifted-Legendre Galerkin algorithm

In this research, we investigate Hilfer-type FDDEs of both linear and nonlinear nature, with orders α and β. We studied some Riemann-Liouville, Caputo-Liouville, and Hilfer fractional derivatives properties; and then used them to convert our fractional delay problem into an equivalent fractional Volterra delay problem. After that, for the resulting fractional Volterra delay problem, using the CMT, existence-uniqueness is demonstrated. For numerical solutions, we studied the orthogonal shifted Legendre polynomials and used them as the basis for the Galerkin algorithm to find an approximation solution to the corresponding FDDE. By employing this algorithm, the provided FDDE is converted into a series of algebraic systems. By solving this system, the approximated solutions for the equation are obtained. The main algorithm advantage is that the number of needed iterations is low compared with the soft gained results. Some linear and nonlinear numerical applications together with several figures and tables are utilized to prove the performance and accuracy of the algorithm. The outcomes results are considered according to the given analysis and the numerical method was offered in the last section with several hints to guide future work.