In recent years, fractional-order partial differential equation models have been proposed and investigated in many research fields, such as fluid mechanics, mechanics of materials, biology, plasma physics, finance, chemistry and so on. Fractional-order differential equations, such as fractional Fokker-Plank equation, fractional nonlinear Schr¨odinger equation, fractional Navier-Stokes equation, fractional quasi-geostrophic equation, fractional Ginzburg-Landau equation and fractional Landau-Lifshitz equation have clear physical background and opened up related new research fields. In fact, some mathematicians (such as L’Hˆopital, Leibniz, Euler) began to consider how to define the fractional derivative as early as the end of the 17th century.