Uniqueness and numerical reconstruction for inverse problems dealing wit interval size search

TL;DR

This paper investigates inverse problems for heat and wave equations in one-dimensional intervals, establishing uniqueness, non-uniqueness, and size estimates, along with numerical methods for accurate interval size reconstruction.

Contribution

It provides new theoretical results on uniqueness and size estimates for inverse interval problems and introduces numerical algorithms for accurate size reconstruction.

Findings

Proved conditions for uniqueness and non-uniqueness.

Developed numerical methods for size approximation.

Validated methods with numerical experiments.

Abstract

We consider a heat equation and a wave equation in a spatial interval over a time interval. This article deals with inverse problems of determining sizes of spatial intervals by extra boundary data of solutions of the governing equations. Under several different circumstances, we prove the uniqueness, the non-uniqueness and some size estimate. Moreover, we numerically solve the inverse problems and compute accurate approximations of the sizes. This is illustrated with satisfactory numerical experiments.