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May  2020, 13(5): 1589-1599. doi: 10.3934/dcdss.2020090

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

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

Received  October 2017 Revised  April 2018 Published  June 2019

Fund Project: 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)

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.

Citation: Luciano Pandolfi. Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1589-1599. doi: 10.3934/dcdss.2020090
References:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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