I'm working on some papers which use Caputo fractional evolution equation (see on Wikipedia) as application for thier main result:
For example:
$$\left\{\begin{matrix} ^CD^{\sigma}_tx(t)+Ax(t)=&f(t,x(t),\int_{0}^{t}k(t,s,x(s))ds)\:\:t\in J:=[0,b], \\ x(0)=x_0& \end{matrix}\right.$$
where $^CD^{\sigma}_t$ denotes the Caputo fractional derivative of order $\sigma\in (0,1), $ $-A:D(A)\subset X\rightarrow X$ generates a positive $C_0$-semigroup, and $f$, $k$ are given functions.
I'm wondering is this derivative important than the usual derivative, and why?
Can we provide a geometrical and physical interpretations for fractional derivatives like as integer order derivatives?
PS: I need a good book or article about Caputo fractional derivative !