Identification of a relaxation kernel using two boundary measures

TL;DR

This paper presents a method to identify a relaxation kernel in systems with memory effects, such as viscoelastic materials, by using two boundary measurements to formulate a linear integral equation for the kernel.

Contribution

It demonstrates that two boundary measures can reduce the nonlinear kernel identification problem to a linear Volterra integral equation, simplifying the process.

Findings

Two boundary measures lead to a linear integral equation for the kernel.

The kernel identification problem becomes mildly ill-posed but linear.

The approach applies to systems with persistent memory like viscoelasticity.

Abstract

We consider a distributed system with persistent memory of a type which is often encountered in viscoelasticity or in the study of diffusion processes with memory. The relaxation kernel, i.e. the kernel of the memory term, is scarcely known from first principles, and it has to be inferred from experiments taken on samples of the material. We prove that two boundary measures give a linear Volterra integral equation of the first kind for the unknown kernel. Hence, with two measures, the identification of the kernel, which in principle is a nonlinear problem, is reduced to the solution of a mildly ill posed but linear problem.