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

A method recently advanced as the conformable Euler method (CEM) for the finite difference discretization of fractional initial value problem Dαty(t)=f(t;y(t)),y(t0)=y0,a≤t≤b, and used to describe hyperchaos in a financial market model, is shown to be valid only for α=1. The property of the conformable fractional derivative (CFD) used to show this limitation of the method is used, together with the integer definition of the derivative, to derive a modified conformable Euler method for the initial value problem considered. A method of constructing generalized derivatives from the solution of the non-integer relaxation equation is used to motivate an alternate definition of the CFD and justify alternative generalizations of the Euler method to the CFD. The conformable relaxation equation is used in numerical experiments to assess the performance of the CEM in comparison to that of the alternative methods.