Fractional crossover delay differential equations of Mittag-Leffler kernel: Existence, uniqueness, and numerical solutions using the Galerkin algorithm based on shifted Legendre polynomials

In the present work, we consider a class of fractional delay differential equations of order p with Atangana-Baleanu fractional derivatives in the Caputo sense. We convert our fractional delay problem to Volterra inte-gral equation and used them to establish the local existence theorem using the Arzela-Ascoli theorem and Schauder's fixed point theorem. After that, the contraction mapping theorem was used to prove the global ex-istence and uniqueness theorem. For a numerical solution, we develop the Galerkin algorithm based on shifted Legendre polynomials to solve the utilized fractional delay problem. This algorithm is based on reducing such problems to those of solving a system of algebraic equations, also we state and prove the convergence and error estimate theorems. Four numerical examples, two linear and two nonlinear are presented to test the efficiency and accuracy of the proposed algorithm with some tables and figures to compare our results with the exact solutions. Several, conclusions, recommendations, inductions, and highlights were formulated in the last chapter to extinguish the perfection of the work.