Analytical and Numerical Analysis for a Nonlinear Fractional Differential Equations Involving the New Caputo–Fabrizio Integral

In this paper we consider a nonlinear fractional differential equations involving the new Caputo–Fabrizio derivative of order $$\gamma \in \;]{\kern 1pt} {\mathbf{1}},{\mathbf{2}}{\kern 1pt} [$$ . We convert the fractional problem to an equivalent nonlinear Volterra integro-differential equation of the second kind, then we investigate the existence and uniqueness of its solution under certain given conditions by using the Schauder fixed point theorem. Finally, we numerically solve the proposed fractional problem by applying the Nyström method, and we provide some suitable examples to support our study.