A linear algorithm for the identification of a weakly singular relaxation kernel using two boundary measurements

TL;DR

This paper presents a linear algorithm that uses two boundary measurements to identify a relaxation kernel in systems with memory, simplifying the process to a linear deconvolution problem, relevant in diffusion and viscoelasticity.

Contribution

It introduces a novel linear method for identifying Abel-type relaxation kernels from boundary data, reducing the problem to a linear deconvolution.

Findings

Identification reduces to a linear deconvolution problem.

Method applicable to diffusion and viscoelastic systems.

Efficient boundary measurement-based kernel recovery.

Abstract

We consider a distributed system of a type which is encountered in the study of diffusion processes with memory and in viscoelasticity. The key feature of such system is the persistence in the future of the past actions due the memory described via a certain relaxation kernel, see below. The parameters of the kernel have to be inferred from experimental measurements. Our main result in this paper is that by using two boundary measurements the identification of a relaxation kernel which is linear combination of Abel kernels (as often assumed in applications) can be reduced to the solution of a (linear) deconvolution problem.