Fractional Derivative Operator of Multi-Order

Question

(1) Is there any existence of fractional integral or fractional derivatives operator which contains two fractional orders at the same time. I mean if $D^{\alpha}_{\beta}$ is an operator with $\alpha$ and $\beta$ are its fractional orders and we discuss the fractional derivative by taking different values of $\alpha$ and $\beta$ at the same time. (2) I have defined such multi-order fractional operators but don't know where are they applicable?. Please guide me by writing the names of those fields where such operators can be applied. Thank you very much.

Answer 1

I suppose that your question doesn't concern the fractional derivatives of order $\alpha$ and $\beta$ successively applied to the same function of only one variable : $$D^{\alpha+\beta}f(x)=D^{\beta}\left(D^{\alpha}f(x)\right)=D^{\alpha}\left(D^{\beta}f(x)\right)$$ I suppose that, in your sense, "two fractional orders at the same time" means something similar to the usual partial derivatives, but in the field of fractional derivatives, i.e.: "Partial fractional derivatives".

Nothing prevents to compute the fractional derivative of a function of several variables.

For more generality, with the Caputo's definition of fractional derivative of order $\nu$ and with $n=\lfloor \nu \rfloor$ : $$ _aD_x^\nu f(x)=\frac{1}{\Gamma(n-\nu)}\int_a^x\frac{\frac{d^nf(t)}{dt^n}}{(x-t)^{\nu-n+1}}dt$$

Of course, other variants of definitions where proposed : https://en.wikipedia.org/wiki/Fractional_calculus#Fractional_derivatives

Generalization in case of a two variables function $f(x,y)$ : $$ _{a,b}D_{x,y}^{\nu,\mu} f(x,y)=\frac{1}{\Gamma(n-\nu)\Gamma(m-\mu) }\int_b^y \int_a^x \frac{\frac{\partial^{n+m}f(t,\tau)}{\partial t^n\partial \tau^m}}{(x-t)^{\nu-n+1} (y-\tau)^{\mu-m+1}}dt\,d\tau\: $$ where $n=\lfloor \nu \rfloor$ and $m=\lfloor \mu \rfloor$

With respect to the analogy to the classical partial derivatives, another symbol could be proposed : $$ _{a,b}\boldsymbol{\partial}_{x,y}^{\nu,\mu} f(x,y).$$ I would imagine that the applications could generalize the known cases of applications, such as a lot are mentioned for example in the book : Keith B.Oldham, Jerome Spanier, The Fractional Calculus, Academic Press, New York, 1974.

More specific, in the field of equivalent electrical networks models, for example in the paper : https://fr.scribd.com/doc/71923015/The-Phasance-Concept , a lot of networks models are considered represented in two dimensions (drawn on a flat sheet). One can imagine networks in three dimensions (so, in volume) in order to model electrical phenomena involving not only one so called "phasance" property of degree $\nu_1$, but several $\nu_2$,... in a common space.

One can found on the web several papers with key words "partial fractional differential equations".

Answer 2

Thank you very much @JJacquelin for your so explanatory comments. But my confusion remains constant. Why do you suppose this a partial fractional derivatives?. Let me explain my question once again. As there are several approaches exist in the literature for the definitions of fractional derivatives. For example Riemann-Liouville approach, Grunwald-Letnikov approach, caputo definition and so on. I take a fractional derivative operator $A_{\alpha}$ of order $\alpha$ and apply an approach Riemann-Liouville or Grunwald or caputo etc. As a result i get an another fractional derivative operator $D^{\alpha}_{\beta}$ which contains two fractional orders $\alpha$ and $\beta$ at the same time. This operator can be applied to a single variable function $f(x)$, and we can discuss fractional derivatives for different values of $\alpha$ and $\beta$ at the same time. For your information here the parameter $\alpha$ already existed, which is the fractional order of the operator $A_{\alpha}$, while the second parameter $\beta$ appears during construction (applying a fractional derivative approach to $A_{\alpha}$). what do you say about this? Is this work has some applications?