The nonlocal coupled system of Caputo-Fabrizio fractional q-integro differential equation

This scheme's main goal is to examine the existence, uniqueness, and continuous dependence of solutions for a nonlinear coupled system of fractional q-integro-differential equations involving the derivation and integration of fractional Caputo-Fabrizio. The numerical technique methodology of the proposed problem will be introduced. Proving the existence theorem depends on Schauder's fixed-point theorem. To drive the numerical method, we use the definitions of the fractional derivative and integral of Caputo-Fabrizio and the q-integral of the Riemann-Liouville type. Then, the integral part will be treated using the trapezoidal method, and the derivative part will be treated using the forward finite difference method. And therefore, the coupled system will be converted into a system of algebraic equation that will be solved together to get the solutions. Finally, we give two examples to illustrate the effectiveness of the suggested approach.