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?

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systems, it is necessary to use fractal operators because the separation of time-scales of classical physics is no longer valid.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.
