On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

Abstract

Abstract:It has been pointed out that the derivative chains rules in fractional differential calculus via fractional calculus are not quite satisfactory as far as they can yield different results which depend upon how the formula is applied, that is to say depending upon where is the considered function and where is the function of function. The purpose of the present short note is to display some comments (which might be clarifying to some readers) on the matter. This feature is basically related to the non-commutativity of fractional derivative on the one hand, and furthermore, it is very close to the physical significance of the systems under consideration on the other hand, in such a manner that everything is right so. As an example, it is shown that the trivial first order system may have several fractional modelling depending upon the way by which it is observed. This suggests some rules to construct the fractional models of standard dynamical systems, in as meaningful a model as possible. It might happen that this pitfall comes from the feature that a function which is continuous everywhere, but is nowhere differentiable, exhibits random-like features. PACS

On chain rule for fractional derivatives

“…Authors of some papers ( [11]- [18] and [24]- [28]) suggest new types of fractional derivatives and assume some usual or simplified rules for the suggested derivatives. For example, the chain rule, which is suggested in [11]- [16], has the form…”

Two Reliable Methods for Solving the (3 + 1)‐Dimensional Space‐Time Fractional Jimbo‐Miwa Equation

“…Jumarie's modified Riemann-Liouville derivative of order is defined by the following expression [53,54]:…”

“…Properties (7)- (9) are direct results [53][54][55] from using the fractional Leibniz rule, the fractional Barrow's formula, and the relation ( ) ≅ Γ( + 1) ( ). Property Mathematical Problems in Engineering 3 (7) requires only that the functions and are continuous (not necessarily differentiable) functions.…”

“…However, their work did not produce the exact solutions for Cases 1 and 3, that is, (54)- (80) and (108)-(134). In addition, our exact solutions in (54), (66), (105), and (107) are equivalent to that, which were constructed by the exp(−Φ( )) method, in [66] in which they were expressed in (28), (29), (30), and (32), respectively. Furthermore, our exact solutions in (79) and (106) have the same mathematical structure as solution (29) in [67] using the exp-function method.…”

Sliding Mode Controller Design Based on Backstepping Technique for Fractional Order System

“…G. Jumarie (2013) displayed the chain rule of fractional differentiation based on analytical solution of fractional order derivative of power functions [8].…”