Determination of the body force of a two-dimensional isotropic elastic body

TL;DR

This paper addresses the inverse problem of determining the body force in a 2D isotropic elastic body using Fourier transform and regularization techniques, providing explicit error estimates and numerical validation.

Contribution

It introduces a regularization method for a nonlinear, ill-posed inverse problem with inexact data, including explicit error bounds and numerical results.

Findings

Explicit error estimates for the regularized solution

Effective numerical implementation demonstrated

Regularization stabilizes the solution of the nonlinear inverse problem

Abstract

Let $\Omega$ represent a two$-$dimensional isotropic elastic body. We consider the problem of determining the body force $F$ whose form $\phi(t)(f_1(x),f_2(x))$ with $\phi$ be given inexactly. The problem is nonlinear and ill-posed. Using the Fourier transform, the methods of Tikhonov's regularization and truncated integration, we construct a regularized solution from the data given inexactly and derive the explicitly error estimate. Numerical part is given