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

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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  • 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.

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


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  • [1] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory. Theory and Applications,, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.
    [2] S. Avdonin and L. Pandolfi, A linear algorithm for the identification of a weakly singular relaxation kernel using two boundary measurements, J. Inv. Ill Posed Problems, 26 (2018), 299-310.  doi: 10.1515/jiip-2016-0064.
    [3] A. Belleni-Morante, An integro-differential equation arising from the theory of heat conduction in rigid materials with memory, Boll. Unione Mat. Ital., 15 (1978), 470-482. 
    [4] D. Brandon and W. J. Hrusa, Construction of a class of integral models for heat flow in materials with memory, J. Integral Equations Appl., 1 (1988), 175-201.  doi: 10.1216/JIE-1988-1-2-175.
    [5] D. L. BykovA. V. KazakovD. N. KonovalovV. P. Me'lnikovA. N. Osavchuk and V. A. Peleshko, Identification of the model of nonlinear viscoelasticity of filled polymer materials in millisecond time range, Mechanics of Solids, 47 (2012), 641-645.  doi: 10.3103/S0025654412060052.
    [6] F. ColomboD. Guidetti and V. Vespri, Identification of two memory kernels and the time dependence of the heat source for a parabolic conserved phase-field model, Math. Methods Appl. Sci., 28 (2005), 2085-2115.  doi: 10.1002/mma.658.
    [7] W. A. Day, On monotonicity of the relaxation functions of viscoelastic materials, Proc. Camb. Phil. Soc., 67 (1970), 503-508.  doi: 10.1017/S0305004100045771.
    [8] V. P. GolubB. P. Maslov and P. V. Fernati, Identification of the hereditary kernels of isotropic linear viscoelastic materials in combined stress state. 1. Superposition of shear and bulk kreep,, International Applied mechanics, 52 (2016), 648-660.  doi: 10.1007/s10778-016-0744-8.
    [9] D. Guidetti, Reconstruction of a bounded variation convolution kernel in an abstract wave equation, Forum Math., 22 (2010), 1129-1160.  doi: 10.1515/FORUM.2010.060.
    [10] M. E. Gurtin and A. G. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Rational Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373.
    [11] J. Janno and A. Lorenzi, Recovering memory kernels in parabolic transmission problems, J. Inverse Ill-Posed Probl., 16 (2008), 239-265.  doi: 10.1515/JIIP.2008.015.
    [12] A. Lorenzi and E. Rocca, Identification of two memory kernels in a fully hyperbolic phase-field system, J. Inverse Ill-Posed Probl., 16 (2008), 147-174.  doi: 10.1515/JIIP.2008.010.
    [13] A. Lorenzi and E. Sinestrari, An inverse problem in the theory of materials with memory, Nonlinear Anal., 12 (1988), 1317-1335.  doi: 10.1016/0362-546X(88)90080-6.
    [14] E. Pais and J. Janno, Identification of two degenerate time-and space-dependent kernels in a parabolic equation, Electron. J. Differential Equations, 180 (2005), 1-20. 
    [15] L. Pandolfi, Riesz systems and the controllability of heat equations with memory,, Int. Eq. Operator Theory, 64 (2009), 429-453.  doi: 10.1007/s00020-009-1682-1.
    [16] L. Pandolfi, Distributed Systems with Persistent Memory. Control and Moment Problems,, Springer Briefs in Electrical and Computer Engineering. Control, Automation and Robotics. Springer, Cham, 2014. doi: 10.1007/978-3-319-12247-2.
    [17] L. Pandolfi, A linear algorithm for the identification of a relaxation kernel using two boundary measures,, Inverse Problems, 31 (2015), 105003, 12 pp. doi: 10.1088/0266-5611/31/10/105003.
    [18] L. Pandolfi, Identification of the relaxation kernel in diffusion processes and viscoelasticity with memory via deconvolution,, Math. Methods Appl. Sci., 40 (2017), 2542-2549.  doi: 10.1002/mma.4180.
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