existence and numerical simulation of solutions for a caputo-fabrizio fractional differential equation

The present research investigates the existence and numerical simulation of solutions for a class of caputo-fabrizio fractional differential equations. at first we obtain a prior estimate for solutions of a functional integral equation that is related to the main problem. then using a fixed point theorem, the existence of at least one smooth  solution is proved. furthermore, a new numerical method based on b-spline is developed to approximate the solution. it is proved that  a locally superconvergent approximation is achieved via even-degree splines on the mid points of the uniform partition. the convergence of the proposed method is analyzed using an operator-based approach, and the corresponding theoretical convergence orders are rigorously derived.  finally, several illustrative examples are presented to demonstrate the efficiency and applicability of the method. the results of the numerical experiments confirm the theoretical predictions concerning the convergence orders.