Some inverse problems for wave equations with fractional derivative attenuation

Abstract

The damped wave equation with the attenuation proportional to velocity is ubiquitous in science and engineering and a common situation is when the attenuation depends on frequency. The usual way to incorporate this effect is to introduce fractional order derivatives either as a replacement for u t or as modifier through a spatial component with space fractional derivatives. Models for these are very well developed and the effort in this paper is towards the analysis of the inverse problem of recovering critical coefficients or initial states although we also develop constructive methods for these and analyse their degree of ill-conditioning.