Initial Boundary Value and Inverse Coefficient Problems for One-Dimensional Fractional Diffusion Equation in a Half-Line

In this paper, we propose new formulas for representing the solution of the first (Dirichlet), second (Neumann) and third (Robin) initial-boundary value problems for one-dimensional fractional diffusion equations with the time-fractional Caputo derivative. These formulas are obtained by the continuation method used in the theory of differential equations with integer derivatives. The Green’s functions of the problems are constructed in terms of the Fox \(H\) -function. As an example of the use of the constructed formulas, the inverse problem of determining the zero-coefficient of the fractional diffusion equation with the Neumann boundary condition and the overdetermination condition at the boundary of the spatial domain is studied. The existence and uniqueness of the solution of the considered inverse problem is obtained by a method based on the integral equations using Green’s function. It is shown that this solution depends continuously on the input data.