# Fractional Derivative Operator of Multi-Order

(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.

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".

• I repeat my question in another way, as under @JJacquelin – Tahir Ullah Khan Jun 5 '17 at 10:32

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?

• If I well understand what you say, the combination of a first operator $A_\alpha$ and then a second operator $A_\beta$ is nothing else that the operator of fractional derivation of order $\alpha+\beta$. There is nothing new. Sorry if my understanding of your idea is not correct. To avoid confusion, please edit the mathematical definition of your symbolism $D_\beta^\alpha$ on explicit form of integral(s). – JJacquelin Jun 5 '17 at 10:51