Abstract
A new modified numerical scheme is proposed. This idea is inspired through the general idea of the Lagrange and the Adams-Bashforth. We present the derivation of the general steps for a modified numerical scheme for an equation with two different fractional derivatives. We present an application of such method in a general nonlinear Cauchy problem. The application is presented to some fractional differential equations. Additionally we realized that when the non-integer order is one, the well-known multi-step-Adams-Bashforth numerical scheme for classical differential equations is recovered. (C) 2019 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.