Some Inverse Problems for the Burgers Equation and Related Systems

TL;DR

This paper investigates inverse problems for the Burgers equation and related systems, focusing on determining spatial interval sizes from boundary data, with results on uniqueness, non-uniqueness, and numerical approximations.

Contribution

It provides new theoretical results on uniqueness and non-uniqueness, and develops numerical methods for estimating interval sizes in inverse problems for nonlinear PDEs.

Findings

Proved conditions for uniqueness and non-uniqueness.

Developed numerical algorithms for interval size approximation.

Validated methods through computational experiments.

Abstract

In this article we deal with one-dimensional inverse problems concerning the Burgers equation and some related nonlinear systems (involving heat effects and/or variable density). In these problems, the goal is to find the size of the spatial interval from some appropriate boundary observations of the solution. Depending on the properties of the initial and boundary data, we prove uniqueness and non-uniqueness results. In addition, we also solve some of these inverse problems numerically and compute approximations of the interval sizes.