# The Problem of Finding the One-Dimensional Kernel of the Thermoviscoelasticity Equation

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## Abstract

The problem of determining the kernel *h*(*t*), *t* ∈ [0, *T*], appearing in the system of integro-differential thermoviscoelasticity equations is considered. It is assumed that the coefficients of the equations depend only on one space variable. The inverse problem is replaced by the equivalent system of integral equations for unknown functions. The contraction mapping principle with weighted norms is applied to this system in the space of continuous functions. A global unique solvability theorem is proved and an estimate of the stability of the solution of the inverse problem is obtained.

## Keywords

inverse problem stability delta function Lame´ coefficients kernel## Preview

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## References

- 1.M. Grasselli, S. I. Kabanikhin, and A. Lorenzi, “An inverse problem for an integro-differential equation,” Sibirsk. Mat. Zh.
**33**(1), 58–68 (1992) [Sib. Math. J. 33 (1), 415–426 (1992)].MathSciNetMATHGoogle Scholar - 2.A. Lorenzi and E. Paparoni, “Direct and inverse problems in the theory of materials with memory,” Rend. Sem. Mat. Univ. Padova
**87**, 105–138 (1992).MathSciNetMATHGoogle Scholar - 3.A. Lorenzi and V. I. Priimenko, “Identification problems related to electro-magneto-elastic interactions,” J. Inverse Ill-Posed Probl.
**4**(2), 115–143 (1996).MathSciNetCrossRefMATHGoogle Scholar - 4.J. Janno and L. Von Wolfersdorf, “Inverse problems for identification memory kernels in viscoelasticity,” Math. Methods Appl. Sci.
**20**(4), 291–314 (1997).MathSciNetCrossRefMATHGoogle Scholar - 5.A. L. Bukhgeim, N. I. Kalinina, and V. B. Kardakov, “Two methods for the inverse problem of memory reconstruction,” Sibirsk. Mat. Zh.
**41**(4), 767–776 (2000) [Sib. Math. J. 41 (4), 634–642 (2000)].MathSciNetMATHGoogle Scholar - 6.D. K. Durdiev and Zh. D. Totieva, “The problem of determining the one-dimensional kernel of the viscoelasticity equation,” Sib. Zh. Ind. Mat.
**16**(2), 72–82 (2013).MathSciNetMATHGoogle Scholar - 7.D. K. Durdiev, “An inverse problem for determining two coefficients in an integrodifferential wave equation,” Sib. Zh. Ind. Mat.
**12**(3), 28–40 (2009).MathSciNetMATHGoogle Scholar - 8.D. K. Durdiev and Zh. Sh. Safarov, “Inverse problem of determining the one-dimensional kernel of the viscoelasticity equation in a bounded domain,” Mat. Zametki
**97**(6), 855–867 (2015) [Math. Notes 97 (5–6), 867–877 (2015)].MathSciNetCrossRefMATHGoogle Scholar - 9.V. G. Romanov, “Inverse problems for differential equations with memory,” Eurasian J. Math. and Comput. Appl.
**2**(4), 51–80 (2014).Google Scholar - 10.V. G. Romanov, “Stability estimates for the solution in the problem of determining the kernel of the viscoelasticity equation,” Sib. Zh. Ind. Mat.
**15**(1), 86–98 (2012).MathSciNetMATHGoogle Scholar - 11.V. G. Romanov, “A two-dimensional inverse problem for the viscoelasticity equation,” Sibirsk. Mat. Zh.
**53**(6), 1401–1412 (2012) [Sib. Math. J. 53 (6), 1128–1138 (2012)].MathSciNetGoogle Scholar - 12.A. Lorenzi and V. G. Romanov, “Recovering two Lamékernels in a viscoelastic system,” Inverse Probl. Imaging
**5**(2), 431–464 (2011).MathSciNetCrossRefMATHGoogle Scholar - 13.D. K. Durdiev and Zh. D. Totieva, “The problem of determining the multidimensional kernel of the viscoelasticity equation,” Vladikavkaz. Mat. Zh.
**17**(4), 18–43 (2015).MathSciNetMATHGoogle Scholar - 14.D. K. Durdiev,
*Inverse Problems forMedia with Aftereffects*(Turon–Ikbol, Tashkent, 2014) [in Russian].Google Scholar - 15.V. G. Romanov,
*Stability in Inverse Problems*(Nauchn. Mir, Moscow, 2005) [in Russian].MATHGoogle Scholar - 16.D. K. Durdiev, “Inverse problem for the system of thermoelasticity equations for a vertically inhomogeneous cohesionless medium with memory,” Differ. Uravn.
**45**(9), 1229–1236 (2009) [Differ. Equations 45 (9), 1254–1261 (2009)].MathSciNetMATHGoogle Scholar - 17.A. D. Kovalenko,
*Thermoelasticity*(Vishcha Shkola, Kiev, 1975) [in Russian].Google Scholar - 18.Zh. D. Tuaeva, “Multidimensional Mathematical Seismic Model with Memory,” in
*Studies in Differential Equations and Mathematical Modeling*(VNTs RAN, Vladikavkaz, 2008), pp. 297–306 [in Russian].Google Scholar - 19.V. G. Yakhno,
*Inverse Problems for Differential Equations of Elasticity*(Nauka, Novosibirsk, 1990) [in Russian].MATHGoogle Scholar

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