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# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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

Sergei Avdonin
/ Luciano Pandolfi
Published Online: 2018-03-03 | DOI: https://doi.org/10.1515/jiip-2016-0064

## 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 a 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 a linear combination of Abel kernels (as often assumed in applications) can be reduced to the solution of a (linear) deconvolution problem.

Keywords: 45D05; 45K05; 45Q05

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## About the article

Accepted: 2018-01-23

Published Online: 2018-03-03

Published in Print: 2018-04-01

This work has been done in the context of a visit of the first author to the Dipartimento di Scienze Matematiche “G. L. Lagrange” of the Politecnico di Torino in June 2016, supported by GNAMPA-INDAM. It fits into the research programs of GNAMPA-INDAM and of the “Groupement de Recherche en Contrôle des EDP entre la France et l’Italie (CONEDP-CNRS)”. The research of the first author was supported in part by the National Science Foundation, grant no. DMS 1411564, and by the Ministry of Education and Science of Republic of Kazakhstan under the grant no. AP05136197.

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 26, Issue 2, Pages 299–310, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

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