The computational orthogonal shifted Legendre–Galerkin approach for handling fractional delay differential problems via adapting fractional M-derivative

This paper presents a numerical procedure for handling delay fractional differential problems where the derivative is defined using the M-fractional approach. The proposed scheme modus operandi is based on the shifted Legendre–Galerkin procedure, which is a powerful tool for solving complex differential models of generalized fractional derivatives. The method involves constructing a series of Legendre polynomials that form the basis functions for approximating the solution of the required problem. The coefficients of the series are obtained after solving an algebraic system of linear types that results from the application of the Galerkin practice. The numerical accuracy and convergence assessment are also presented together with various results. Simulations-based analyses are realized to validate the truthfulness and exactness of the process. The results manifest that the M-derivatives and the Galerkin practice provide alternative innovative approaches for handling M-delay fractional problems. Several keynotes and future recommendations are exhibited at the last with some selected references.