Simultaneous determination of coefficients, internal sources and an obstacle of a diffusion equation from a single measurement

TL;DR

This paper addresses the challenging inverse problem of simultaneously identifying coefficients, internal sources, and obstacles in diffusion equations using only a single boundary measurement, applicable to both classical and anomalous diffusion models.

Contribution

It introduces a novel approach for the unique simultaneous determination of multiple unknowns in diffusion equations from minimal boundary data.

Findings

Proves uniqueness of the solution for the inverse problem.

Applies the method to both classical and fractional diffusion equations.

Provides a theoretical framework for practical inverse problem solving.

Abstract

This article is devoted to the simultaneous resolution of three inverse problems, among the most important formulation of inverse problems for partial differential equations, stated for some class of diffusion equations from a single boundary measurement. Namely, we consider the simultaneous unique determination of several class of coefficients, some internal sources (a source term and an initial condition) and an obstacle appearing in a diffusion equation from a single boundary measurement. Our problem can be formulated as the simultaneous determination of information about a diffusion process (velocity field, density of the medium), an obstacle and of the source of diffusion. We consider this problems in the context of a classical diffusion process described by a convection-diffusion equation as well as an anomalous diffusion phenomena described by a time fractional diffusion equation.