Why we use Caputo fractional derivative in application?

Question

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 !

Answer 1

Please see here a chapter from Diethelm, The Analysis of Fractional Differential Equations An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer 2010: https://www.springer.com/cda/content/document/cda_downloaddocument/9783642145735-c1.pdf?SGWID=0-0-45-993489-p174025563

Answer 2

This seems like a good place to start: Properties and Applications of the Caputo Fractional Operator.

For a geometrical/mechanical interpretation, you could look at this MO question and answer.