Uniqueness of simultaneous reconstruction of general space- and time-dependent sources and initial states in fractional diffusion equations and systems from single boundary measurements

Abstract

Inverse problem to determine simultaneously a general space- and time-dependent source and an initial state in a fractional diffusion equation from an {\it a posteriori} measurement of the normal derivative of the state on a portion of a boundary of the space domain is considered. Uniqueness for this problem is proved under the assumption that an order of a fractional derivative involved in the equation is irrational. The uniqueness result is generalized to an inverse problem for a coupled system of fractional diffusion equations, where both sources and initial states are unknown and first component of the state is measured on the boundary.