Exploring the dynamics of nonlocal coupled systems of fractional q-integro-differential equations with infinite delay

In this study, we explore a coupled system of fractional integro-differential equations with infinite delay and nonlocal conditions. This system encompasses classical derivatives of different orders and the fractional derivative of Caputo-Fabrizio type, as well as the fractional integral of the q -Riemann-Liouville operator. We introduce a novel definition of the Caputo and Fabrizio differential operators, enhancing the mathematical formulation. Our main focus is to investigate the system's fundamental properties, including existence, uniqueness, and continuous dependence. Through rigorous mathematical analysis, we establish the existence and uniqueness of solutions and examine how small perturbations in initial conditions or parameters impact the solutions. For the numerical aspect, we use the finite-trapezoidal approach, a reliable method for solving fractional integro-differential equations. We provide a concise explanation of the approach and demonstrate its effectiveness through two numerical examples. Overall, this comprehensive study contributes to the understanding of coupled systems with fractional derivatives and infinite delays, with implications for various scientific and engineering fields.