General solutions to a class of time fractional partial differential equations

A class of time fractional partial differential equations is considered, which includes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier-Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine-Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.