Modified numerical methods for fractional differential equations

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.