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

Abstract

A method recently advanced as the conformable Euler method (CEM) for the finite difference discretization of fractional initial value problem Dtαyt = ft;yt, yt0 = 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.

Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk

“…which has been conclusively shown in [50] to be valid only for α = 1. It is shown in [50] that a modified CEFD (MCEFD) may be obtained from the following alternate CFD definition, which is equivalent to the fractional change of variables in the integer-valued derivative (see also [82]): Definition 1. Given a real-valued function on [0, ∞), the conformable fractional derivative has the following alternative definition:…”

“…Further, as suggested in [13], studies incorporating real economic data with parameter estimation for the financial system with market confidence and ethics for all these derivatives are also necessary. Finally, as can be easily seen from Theorem 4.1 of [50], the discretization methods presented here for CFD systems are easy to implement and are equally applicable to all Caputo type derivatives, and hence, to Riemann-Liouville derivatives through their relationship; hence, they have potential to impact a wide range of fractional derivative applications.…”

“…However, the standard Euler discretization of integer-order systems, such as studied in [13], is known to induce (see, e.g., [48,49]) numerical instabilities and spurious behavior where none exist in the continuous system. Moreover, the CEFD method has recently been shown [50] to be valid only for α = 1 and is, therefore, not a valid fractional method. Nonstandard finite difference (NSFD) models have extensively [48] been shown to eliminate induced chaos; the exact spectral derivative discretization finite difference (ESDDFD) methodology is a novel extension, developed in the context of advection-reaction-diffusion equations [51], of the NSFD method to non-integer derivatives [52].…”

Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense

“…It should be mentioned that there is an exciting recent work on the conformable Euler method for finite difference discretization of FIVPs [ 39 , 40 ] showing that the fractional Taylor expansions in terms of the conformable fractional derivative presented in [ 36 ] is valid for . An alternative definition of the conformable fractional derivative introduced in [ 40 ] based on the exact spectral derivative discretization finite difference method showing that the conformable fractional derivative [ 36 ] is a fractional change of a variable rather that a fractional operator. In view of the results of [*], Definition 6 and Theorem 4 are incorrect, and the RPS results-based thereon can therefore be improved.…”

“…[ 40 ] Given a real-valued function on , the conformable fractional derivative has the following alternative definition: where is understood to mean .…”