What is the physical meaning of the fractional integral and fractional derivative?
And many researchers deal with the fractional boundary value problems, and what is the physical background?
What is the applications of the fractional boundary value problem?
This may not be what your looking for but...
In my line of work I use fractional Poisson process a lot, now these arise from sets of Fractional Differential Equations and the physical meaning behind this is that the waiting times between events is no longer exponentially distributed but instead follows a Mittag-Leffler distribution, this results in waiting times between events that may be much longer than what would normally occur if one was to assume exponential waiting times.
One application, but in that case is fractional differencing, not the fracional derivative, is in time series to model long range dependence. Have a look at https://en.wikipedia.org/wiki/Autoregressive_fractionally_integrated_moving_average
The meaning of fractional calculus for physics has been hard to pin down. Podlubny said it was akin to looking for shadows on the walls from focusing on the geometrical interpretation.
That being said, Bruce J. West has written extensively on the subject. In his fantastic book, The Physics of Fractal Operators, he demonstrates the deep connection between fractional derivatives and fractal geometry. He argues that when modeling chaotic thermodynamic systems, it is necessary to use fractal operators because the separation of time-scales of classical physics is no longer valid.
In systems theory, fractional operators allow us to model the memory of control systems through formalisms such as that of the Volterra Series.
Probably not what you are looking for but interesting:
A lot of restrictions of classical calculus is pointed out by Blas M. Vinagre and YangQuan Chen, starting from the following famous quotes given by the Procrustes Bed :“all people must fit the same bed, if tall, cut the legs and if short, stretch the leg” and second one by the Purloined Letter :“same methods must be always successfully applicable, if no solution is found, there is no solution”. If these two cases are not obvious, then one has to look towards a new paradigm and revisit the origins of classical one. One can see the same history behind the fractional order modeling and control over classical (integer) order.
From book: Bandyopadhyay, Bijnan, and Shyam Kamal. "Essence of fractional order calculus, physical interpretation and applications." Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach. Springer International Publishing, 2015. 1-54.
