The Limited Validity of the Fractional Euler Finite Difference Method and an Alternative Definition of the Caputo Fractional Derivative to Justify Modification of the Method

A method, advanced as the fractional Euler finite difference method (FEFDM), a general method for the finite difference discretization of fractional initial value problems (IVPs) for 0 < α ≤ 1 for the Caputo derivative, is shown to be valid only for α = 1. This is accomplished by establishing, through a recently proposed generalized difference quotient representation of the fractional derivative, that the FEFDM is valid only if a property of the Mittag-Leffler function holds that has only been shown to be valid only for α = 1. It is also shown that the FEFDM is inconsistent with the exact discretization of the IVP for the Caputo fractional relaxation equation. The generalized derivative representation is also used to derive a modified generalized Euler’s method, its nonstandard finite difference alternative, their improved Euler versions, and to recover a recent result by Mainardi relating the Caputo and conformable derivatives.