30
$\begingroup$

I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.

I started with the book The Fractional Calculus by Oldham and Spanier, and it comes as no surprise that I favor the Grünwald-Leitnikov derivative. It seems to me a great definition, because it directly generalizes the basic definition of the derivative $\frac {df} {dx}=\lim_{h \rightarrow 0} \frac {f(x)-f(x-h)} {h}$. And it also produces the integral when $\alpha$ is set to be a negative number.

Another (which I think is the Liouville definition, but I'm not sure) generalizes the property of differentiating an exponential $\frac {d^k} {dx^k} e^{rx} = r^ke^{rx}$ and thus if $\frac {d^\alpha} {dx^\alpha}f(x)=\sum A_ne^{nx}$ then $f(x)=\sum A_n n^\alpha e^{nx}$.

A definition, which is used really often for some reason, is the Caputo derivative. Lot of people find it natural that $\frac {d^{\frac 1 2}} {dx^{\frac 1 2}} [1]=0$, but I think it is "evident" that it should be proportional to $x^{-\frac 1 2}$.

Now comes the actual question. Why are there so many definitions of the fractional derivative? Are some of them "better" than the others in some sense? And lastly, is there a general framework, wherein "functions" of differential operators, maybe more general than (fractional) powers, can be given an explicit meaning?

$\endgroup$
39
$\begingroup$

The reason is that the fractional derivative is not a local operator. The usual derivative is a local derivative in the sense that the value of the derivative at one point only depends on the value of the function in a neighborhood of that point. This is not the case for the fractional derivative and that cannot be due to some general theoretical result due to Peetre.

So the definition depends on the domain of definition of the functions under scrutiny. This is not the same definition if we are looking at functions defined on ${\bf R}$ or on $[0,1]$ or on $[0,\infty)$ and of course the derivative of say $\sin$ is not the same in these three cases. Same for the derivative of the constant function.

Fractional derivatives are a particular example of operators obtained using the functional calculus on some operator space. The result of such operation of course depends on the functional space under consideration, which itself is dictated by the context and the problems at hand.

tl;dr: there is not a best definition and the fractional derivatives do not share the nice local properties of the usual derivative, so beware.

$\endgroup$
  • 5
    $\begingroup$ If one views ordinary and fractional derivatives as densely defined operators on function spaces, then there are almost as many ordinary derivatives as fractional derivatives, as one can vary the function space or norm, alter the domain, impose boundary conditions, etc.. The difference is that all the different possibilities for ordinary differentiation coincide when working locally with nice functions away from boundaries, which is not the case for fractional differentiation. (Also branch cuts can yield some additional variability for fractional derivatives compared with ordinary ones.) $\endgroup$ – Terry Tao Nov 7 '17 at 2:15
2
$\begingroup$

A general framework that gives meaning to functions of differential operators that are not restricted to powers is the theory of pseudodifferential operators.

It is not at all beyond the bounds of possibility that a unifying theory of fractional differintegration will emerge, either within the theory of pseudodifferential operators or outside it. If it does, how it will relate to other functions of the differential operator than powers is not known.

Are some of (the many definitions of the fractional derivative) "better" than the others in some sense?

I'm not sure about this. As well as Riemann-Liouville, Grunwald-Letnikov, and Caputo, the others have included some that have hardly been referred to after they were announced. The hottest definition in the fractional calculus field at the moment - by number of papers that refer to it - is the Atangana-Baleanu one, which I asked about here.

$\endgroup$
0
$\begingroup$

I will give an answer concerning definitions of fractional\nonlocal derivatives that are Markovian generators of stochastic processes with jumps. I will briefly argue that

  • Different definitions arise naturally,
  • there is a clear interpretation of many properties (like nonlocality or killing/not-killing constants), and
  • generalizations are natural and meaningful for applications.

It is useful to look at the most simple stochastic jump process and its corresponding generator. Take a Markov chain $P=\{p_{i,j}\}_{i,j\in \text{State space}}$ (which is intrinsically jumpy) and write out its generator
$$ \mathcal G f(x):=(P-I)f(x)=\sum_{y\in\text{ State space}}(f(y)-f(x))p_{x,y},\quad x\in\text{ State space}. $$ Here the intuition is clear: the infinitesimal jump (working with unit time in this case) from $x$ to $y$ is assigned intensity/probability $p_{x,y}$. The operator $\mathcal G$ is non-local. If we modify the process (impose boundary conditions), say by forcing the process to be absorbed at $a\in\text{ State space}$ once it tries to jump to a state $y\notin \Omega\subset \text{State space},$ we obtain a new generator $$ \mathcal G^{\text{abs}} f(x):=(P^{\text{abs}}-I)f(x)=\sum_{y\in\Omega}(f(y)-f(x))p_{x,y}+(f(a)-f(x))\sum_{y\notin\Omega}p_{x,y},\quad x\in\Omega. $$ If we instead decide to kill it (by testing against functions with $f(a)=0$, for example), the new generator will be $$ \mathcal G^{\text{kill}} f(x):=(P^{\text{kill}}-I)f(x)=\sum_{y\in\Omega}(f(y)-f(x))p_{x,y}-f(x)\sum_{y\notin\Omega}p_{x,y},\quad x\in\Omega. $$ So from one single process we can obtain many different generators/fractional derivative (as mentioned in a comment above, the boundary conditions are reflected in the representation of the operator away from the boundary due to the non-locality of $\mathcal G$).

Let us now move to the Riemann-Liouville and Caputo derivatives of order $\beta\in(0,1)$. Consider the three fractional derivatives for $x<a$ \begin{align} D^{\beta}_{\infty}f(x)&:= \int_0^{\infty}(f(x+y)-f(x))\nu(y)dy, \\ ^{C}D^{\beta}_a f(x):&= \int_0^{a-x}(f(x+y)-f(x))\nu(y)dy &+(f(a)-f(x))\int_{a-x}^\infty\nu(y)dy,\\ ^{RL}D^{\beta}_af(x)&:= \int_0^{a-x}(f(x+y)-f(x))\nu(y)dy &-f(x)\int_{a-x}^\infty\nu(y)dy, \end{align} where $\nu(y):=\frac{-\Gamma(-\beta)^{-1}}{y^{1+\beta}}$. Similarly as for the Markov chain above: the operator $D^{\beta}_{\infty}$ is the generator of a $\beta$-stable subordinator $X^\beta(s)$, the operator $^{C}D^{\beta}_a$ is the generator of a $\beta$-stable subordinator $X^\beta(s)$ absorbed at $\{a\}$ on the first attempt to jump outside $\Omega:=(-\infty,a)$, and the operator $^{RL}D^{\beta}_a$ is the generator of a $\beta$-stable subordinator $X^\beta(s)$ killed on the first attempt to jump outside $\Omega:=(-\infty,a)$. Integrating by parts we can rewrite the three operators above in their Riemann-Liouville integral representation, namely \begin{align} D^{\beta}_{\infty}f(x)&= \int_x^{\infty}f'(y)\frac{(y-x)^{-\beta}}{\Gamma(1-\beta)}dy \\ ^{C}D^{\beta}_a f(x)&= \int_x^{a}f'(y)\frac{(y-x)^{-\beta}}{\Gamma(1-\beta)}dy,\\ ^{RL}D^{\beta}_af(x)&= \frac{d}{dx}\int_x^{a}f(y)\frac{(y-x)^{-\beta}}{\Gamma(1-\beta)}dy, \end{align} where the last two operators are your standard definitions of Caputo and Riemann-Liouvile derivatives (right and left versions will correspond to the processes $X^\beta(s)$ and $-X^{\beta}(s)$ respectively). We can now say that the Caputo derivative $^{C}D^{\beta}_a$ (Riemann-Liouville derivative $^{RL}D^{\beta}_a$) kills (does not kill) constants as it is the generator of a process (killed process). Again you can see that (naturally) $^{C}D^{\beta}_a$ and $^{RL}D^{\beta}_a$ contain boundary information in their representation away from the boundary (in sharp difference with local differential operators). Some references: Caputo, Riemann-Liouville, and Grünwald-Leitnikov derivatives from a stochastic point of view in this book. Reflecting boundary conditions and other options for Caputo derivatives of order $\beta\in(1,2)$ here and here.

By substituting a general Lévy measure $\nu(x,dy)$ in the formulas above (generalizing fractional derivatives), many meaningful stochastic processes and their versions on a bounded domain can be studied through their generators (see book, article ). Similar arguments can be carried over for some fractional Laplacians (see this book for example).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.