Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory

Luciano Pandolfi

doi:10.3934/dcdss.2020090

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2020, Volume 13, Issue 5: 1589-1599. Doi: 10.3934/dcdss.2020090

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Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory

Luciano Pandolfi

Dipartimento di Scienze Matematiche "G.L. Lagrange" (retired), Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy

Received: September 30, 2017

Revised: March 31, 2018

Published: March 2020

This research was partially supported by the Politecnico di Torino, and by the GDRE (Groupement de Recherche Européen) ConEDP (Control of PDEs). The author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

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Abstract

In this paper we present a linear method for the identification of both the energy and flux relaxation kernels in the equation of thermodynamics with memory proposed by M.E. Gurtin and A.G. Pipkin. The method reduces the identification of the two kernels to the solution of two (linear) deconvolution problems. The energy relaxation kernel is reconstructed by means of energy measurements as the solution of a Volterra integral equation of the first kind which does not depend on the still unknown flux relaxation kernel. Then, flux measurements are used to identify the flux relaxation kernel.

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Mathematics Subject Classification: Primary: 45K05; Secondary: 45Q05, 93B05.

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