Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation

TL;DR

This paper establishes the simultaneous uniqueness of identifying multiple parameters, including fractional order, potential, initial values, and boundary coefficients, in a one-dimensional time-fractional diffusion-wave equation using boundary data.

Contribution

It introduces a novel approach combining Laplace transform and transformation formulas to prove parameter uniqueness in fractional diffusion-wave equations.

Findings

Uniqueness in determining fractional order, potential, initial values, and Robin coefficients.

Applicability to inverse problems with input source terms.

Conditions on eigenmodes for the uniqueness result.

Abstract

This article deals with the uniqueness in identifying multiple parameters simultaneously in the one-dimensional time-fractional diffusion-wave equation of fractional time-derivative order $\in (0,2)$ with the zero Robin boundary condition. Using the Laplace transform and a transformation formula, we prove the uniqueness in determining an order of the fractional derivative, a spatially varying potential, initial values and Robin coefficients simultaneously by boundary measurement data, provided that all the eigenmodes of an initial value do not vanish. Furthermore, for another formulation of inverse problem with input source term in place of initial value, by the uniqueness in the case of non-zero initial value and a Duhamel principle, we prove the simultaneous uniqueness in determining multiple parameters for a time-fractional diffusion-wave equation.