Inverse problems for fractional Schrödinger and subdiffusion equations

Abstract

The inverse problems of determining the right-hand side of  the Schrödinger and the sub-diffusion equations with the fractional derivative is considered. In the problem 1, the time-dependent source identification problem for the Schrödinger equation , in a Hilbert space  is investigated. To solve this inverse problem, we take the additional condition  with an arbitrary bounded linear functional . In the problem 2, we consider the subdiffusion equation with a fractional derivative of order , and take the аbstract operator as the elliptic part. The right-hand side of the equation has the form , where  is a given function and the inverse problem of determining element is considered. The condition  is taken as the over-determination condition, where  is some interior point of the considering domain and  is a given element. Obtained results are new even for classical diffusion equations. Existence and uniqueness theorems for the solutions to the problems under consideration are proved.