An analytical approach to reconstruct the initial distribution of a sub-diffusion model with time-fractional derivative

The inverse problem has been widely used in optical design, image processing and many other fields. This study is to employ an analytical approach to reconstruct the initial distribution of the fractional sub-diffusion model with Caputo's definition of fractional derivative in time. The basic strategy is to solve the corresponding direct problem via separation of variables and Laplace transform and then to convert the initial inverse problem into an integral equation of the first kind. The key point is that we employ the Picard's theorem to design an analytical solution of initial diffusion distribution. The proposed scheme is tested to some benchmark problems. Numerical results show that the analytical approach performs efficiently.