On the Cauchy problem for the wave equation in a two-dimensional domain with data on the boundary

TL;DR

This paper investigates the stability of solving the wave equation's Cauchy problem in a 2D domain with boundary data, showing improved stability when data is given on the entire boundary.

Contribution

It adapts a previous algorithm to the case where boundary data is on the entire boundary, demonstrating enhanced stability of the solution.

Findings

Higher stability achieved with boundary data on the entire boundary

Algorithm adaptation for full boundary data

Improved solution determination in the special case

Abstract

The subject of the paper is the Cauchy problem for the wave equation in a space-time cylinder $\Omega\times{\mathbb R}$, $\Omega\subset{\mathbb R}^2$, with the data on the surface $\partial\Omega\times I$, where $I$ is a finite time interval. The algorithm for solving the Cauchy problem with the data on $S\times I$, $S\subset\partial\Omega$, was obtained previously. Here we adapt this algorithm to the special case $S=\partial\Omega$ and show that in this situation, the solution is determined with higher stability in comparison with the case $S\subsetneqq\partial\Omega$.