The body force in a three-dimensional Lame' system: identification and regularization

TL;DR

This paper addresses the inverse problem of identifying the unknown spatial component of a body force in a 3D elastic Lame' system, proposing a regularization method with explicit error estimates based on interpolation and Fourier series.

Contribution

It introduces a regularization approach for the ill-posed inverse problem of determining the body force's spatial part using interpolation and Fourier series techniques.

Findings

Constructed a regularized solution with explicit error bounds.

Demonstrated the effectiveness of the method with approximate data.

Addressed the ill-posedness of the inverse problem in elastic systems.

Abstract

Let a three-dimensional isotropic elastic body be described by the Lam\'e system with the body force of the form $F(x,t)=\phi(t)f(x)$, where $\phi$ is known. We consider the problem of determining the unknown spatial term $f(x)$ of the body force where the surface stress history is given as the overdetermination. This inverse problem is ill-posed. Using the interpolation method and truncated Fourier series, we construct a regularized solution from approximate data and provide explicit error estimates. AMS 2010 Subject Classification: 35L20, 35R30. Keywords: Body force, elastic, ill$-$posed problem, interpolation, Fourier series.