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

TL;DR

This paper proves the uniqueness of reconstructing space- and time-dependent sources and initial states in fractional diffusion equations from boundary measurements, under certain conditions, and extends the results to coupled systems.

Contribution

It establishes the first uniqueness results for simultaneous reconstruction of sources and initial states in fractional diffusion systems from boundary data.

Findings

Uniqueness is proved assuming the fractional derivative order is irrational.

Results are extended to coupled fractional diffusion systems.

Boundary measurements suffice for unique reconstruction under specified conditions.

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.