Regularization of an Ill-posed Cauchy Problem for the Wave Equation (Fourier Method)

TL;DR

This paper introduces a Fourier-based regularization algorithm for solving an ill-posed Cauchy problem associated with the wave equation, providing an explicit formula to stabilize the solution process.

Contribution

It presents a novel Fourier method to regularize an ill-posed wave equation problem with explicit formulas, advancing solution stability.

Findings

Developed an explicit regularization formula using Fourier methods.

Provided a stable solution approach for the ill-posed Cauchy problem.

Enhanced understanding of wave equation boundary data reconstruction.

Abstract

An ill-posed Cauchy problem for the wave equation is considered: the solution is to be determined by the Cauchy data on some part of the time-space boundary. By means of Fourier method we obtain a regularization algorithm for this problem, which is given by rather explicit formula.