Numerical analysis of a new volterra integro-differential equation involving fractal-fractional operators

In this paper, we suggest a new integro-differential equation by utilizing from the concept of fractal-fractional derivative and integral newly introduced by Atangana. We offer that under which conditions the solution of the suggested equation is exist and unique benefitting from Banach fixed-point theorem. Also, we construct a new numerical scheme for the numerical solution of our problem and we give numerical simulation and illustrations for different values of fractional order alpha and fractal order beta. That's why, this study will guide for researchers significantly in theory and applications. (C) 2019 Elsevier Ltd. All rights reserved.