What is the physical meaning of fractional calculus? 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.
