Fundamental studies for time fractional partial differential equations and inverse problems.

Fractional derivatives have been studied since Leibniz, and recently by various applications such as diffusion of contaminants in the inhomogeneous media, fractional diffusion equations have drawn great attention. However the serious mathematically fundamental studies for fractional partial equations have just started.  Here I present the formulation of the initial-boundary value problem and the unique existence and related qualitative properties of solutions.  I intend to construct the theory for fractional partial differential equations and  I show a part of it. On the basis of such results, we  discuss an optimal control problem and several inverse problems.