What is( are) the advantage(s) of caputo's to Riemann-Liouville derivation? I am new in fractional calculus. I see most of articles uses Caputo's derivation instead of Riemann-Liouville derivation.


*

*Is there some advantage?

*Can someone make some basic (simple) example for both of them, to help for better understanding.


Any help will be appreciated.
 A: Quote from On Riemann-Liouville and Caputo Derivatives:

In the realm of the fractional differential equations, Caputo derivative and Riemann-Liouville ones are mostly used. It seems that the former is more welcome since the initial value of fractional differential equation with Caputo derivative is the same as that of integer differential equation;

But...

Most people think that these fractional order initial values are not easy to measure. This makes an illusion; that is, RL derivative seems to be used in less situations. But in reality, this is not the case. Physical and geometric interpretations for RL derivative can be found in "An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations". It makes it possible to observe and/or measure values of RL integral and derivatives.

See the examples in the linked article. More examples of RL derivatives in Fractional Calculus: Definitions and Applications.
Also interesting, see the section Riemann versus Caputo in Introduction to fractional calculus.
A: From a probabilistic point of view, when the order of integration is $\beta\in(0,1)$, the small difference between the Riemann-Liouville derivative $^{RL}D^{\beta}_a$ and the Caputo derivative $^{C}D^{\beta}_a$ is that 


*

*$^{RL}D^{\beta}_a$  is the generator of the increasing $\beta$-stable Lévy stochastic process $X^{\beta}(s)$ killed once it tries to cross the barrier $\{a\}$, and

*$^{C}D^{\beta}_a$   is the generator of the the process $X^{\beta}(s)$ absorbed once crossing the barrier $\{a\}$. 


To see this write the two derivatives in their generator forms (which can be obtain by integrating by parts the definitions involving the Riemann-Liouville integrals)
\begin{align}
^{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,\\
^{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,
\end{align}
where $\nu(y):=\frac{-\Gamma(-\beta)^{-1}}{y^{1+\beta}}$, and $x<a$. The intuition is now clear: 

consider the process $X^{\beta}$ starting at $x<a$, then the first term (common to both operators) sums up all the intensities $\nu(y)$ for every jump form $x$ to $y$ falling below $a$ ($0\le y\le a-x$) , indeed
  $$\text{sum}_y\ (f(x+y)-f(x)) \nu(y);$$
  The second term in the Rieman-Liouville case is a standard killing term $-f(x)$ times a (unbounded) coefficient $b(x):=\int_{a-x}^{\infty}\nu(y)dy,$ which contains all the intensity of the jumps that would have fallen above $a$ (for the process starting at $x$). 
  The second term in the Caputo case kills the process with the intensity of all jumps falling above $a$ ( indeed again $(-f(x)b(x))$, BUT also regenerates the process at $a$ through the regenerating term $+f(a)b(x)$ with the same coefficient/sum of the intensities.

A key advantage of Caputo is that it allows to have a nonzero boundary condition $\phi_0$, for instance in time-fractional diffusion equations
$$
^{C}D^{\beta}_0 u= \Delta u, \quad u(0)=\phi_0.
$$
Right or left versions correspond to the increasing $(X^{\beta})$ or the decreasing process $(-X^{\beta})$ with upper or lower barrier ${a}$, respectively. If you are comfortable with Markov chains, take a monotone chain $P$, kill it or absorb it on the attempt of crossing a barrier and try to write out the three (trivial) generators.
For order $\beta\in(1,2)$, for instance you can have a look at this article. Orders above 2 fall out of the Markov processes realm.
