Determining damping terms in fractional wave equations

TL;DR

This paper investigates methods for uniquely identifying multiple fractional damping terms in wave equations, including numerical testing and analysis of the forward problem, with potential extensions to space-dependent coefficients and initial data.

Contribution

It introduces new approaches for the inverse problem of recovering fractional damping terms, utilizing Tauberian theorems and numerical validation.

Findings

Successful reconstruction of damping terms demonstrated numerically

Established uniqueness results for the inverse problem

Analyzed the forward multiterm fractional wave equation

Abstract

This paper deals with the inverse problem of recovering an arbitrary number of fractional damping terms in a wave equation. We develop several approaches on uniqueness and reconstruction, some of them relying on Tauberian theorems on the relation between the asymptotics of solutions in time and Laplace domain. Also the possibility of additionally recovering space dependent coefficients or initial data is discussed. The resulting methods for reconstructing coefficients and fractional orders in these terms are tested numerically. Additionally, we provide an analysis of the forward problem, a multiterm fractional wave equation.