Geodesic
Inverse Problem of Determining the One-Dimensional Kernel of the Viscoelasticity Equation in a Bounded Domain
Matematičeskie zametki, Tome 97 (2015) no. 6, pp. 855-867.

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The one-dimensional integro-differential equation arising in the theory of viscoelasticity with constant density and Lamé coefficients is considered. The direct problem is to determine the displacement function from the initial boundary-value problem for this equation, provided that the initial conditions are zero. The spatial domain is the closed interval $[0,l]$, and the boundary condition is given by the stress function in the form of a concentrated perturbation source at the left endpoint of this interval and as zero at the right endpoint. For the direct problem, we study the inverse problem of determining the kernel appearing in the integral term of the equation. To find it, we introduce an additional condition for the displacement function at $x=0$. The inverse problem is replaced by an equivalent system of integral equations for the unknown functions. The contraction mapping principle is applied to this system in the space of continuous functions with weighted norms. A theorem on the global unique solvability is proved.
Keywords: viscoelasticity equation, integro-differential equation, displacement function, contraction mapping principle, stress function.
Mots-clés : Lamé coefficient 
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     title = {Inverse {Problem} of {Determining} the {One-Dimensional} {Kernel} of the {Viscoelasticity} {Equation} in a {Bounded} {Domain}},
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D. K. Durdiev; Zh. Sh. Safarov. Inverse Problem of Determining the One-Dimensional Kernel of the Viscoelasticity Equation in a Bounded Domain. Matematičeskie zametki, Tome 97 (2015) no. 6, pp. 855-867. http://geodesic.mathdoc.fr/item/MZM_2015_97_6_a4/

[1] A. Lorenzi, E. Paparoni, “Direct and inverse problems in the theory of materials with memory”, Rend. Sem. Mat. Univ. Padova, 1992, no. 87, 105–138 | MR | Zbl

[2] A. Lorenzi, “An identification problem related to a nonlinear hyperbolic integro-differential equation”, Nonlinear Anal.: Theory Methods Appl., 22 (1994), 297–321 | MR | Zbl

[3] A. Lorenzi, V. I. Priimenko, “Identification problems related to electro-magneto-elastic interactions”, J. Inverse Ill-Posed Probl., 4:2 (1996), 115–143 | MR | Zbl

[4] A. L. Bukhgeim, “Inverse problems of memory reconstrution”, J. Inverse Ill-Posed Probl., 1:3 (1993), 193–205 | MR | Zbl

[5] L. Bukhgeim, G. V. Dyatlov, “Edinstvennost v odnoi obratnoi zadache opredeleniya pamyati”, Sib. matem. zhurn., 37:3 (1996), 526–533 | MR | Zbl

[6] A. L. Bukhgeim, N. I. Kalinina, “Globalnaya skhodimost metoda Nyutona v obratnykh zadachakh vosstanovleniya pamyati”, Sib. matem. zhurn., 38:5 (1997), 1018–1033 | MR | Zbl

[7] A. L. Bukhgeim, G. V. Dyatlov, V. M. Isakov, “Ustoichivost vosstanovleniya pamyati po operatoru Dirikhle–Neimana”, Sib. matem. zhurn., 38:4 (1997), 738–749 | MR | Zbl

[8] J. Janno, L. Von Wolfersdorf, “Inverse problems for identification of memory kernels in viscoelasticity”, Math. Methods Appl. Sci., 20:4 (1997), 291–314 | 3.0.CO;2-W class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[9] A. L. Bukhgeim, N. I. Kalinina, V. B. Kardakov, “Dva metoda v obratnoi zadache opredeleniya pamyati”, Sib. matem. zhurn., 41:4 (2000), 767–776 | MR | Zbl

[10] A. Lorenzi, F. Messina, V. G. Romanov, “Recovering a Láme kernel in a viscoelastic system”, Appl. Anal., 86:11 (2007), 1375–1395 | DOI | MR | Zbl

[11] D. K. Durdiev, “Globalnaya razreshimost odnoi obratnoi zadachi dlya integrodifferentsialnogo uravneniya elektrodinamiki”, Differents. uravneniya, 44:7 (2008), 867–873 | MR | Zbl

[12] D. K. Durdiev, “Obratnaya zadacha opredeleniya dvukh koeffitsientov v odnom integrodifferentsialnom volnovom uravnenii”, Sib. zhurn. industr. matem., 12:3 (2009), 28–40 | Zbl

[13] A. Lorenzi, V. G. Romanov, “Recovering two Lamé kernels in a viscoelastic system”, Inverse. Probl. Imaging, 5:2 (2011), 431–464 | DOI | MR | Zbl

[14] D. K. Durdiev, Zh. D. Totieva, “Zadacha ob opredelenii odnomernogo yadra uravneniya vyazkouprugosti”, Sib. zhurn. industr. matem., 16:2 (2013), 72–82 | MR

[15] V. G. Romanov, Obratnye zadachi matematicheskoi fiziki, Nauka, M., 1984 | MR | Zbl

[16] A. S. Alekseev, V. I. Dobrinskii, “Nekotorye voprosy prilozheniya obratnoi dinamicheskoi zadachi seismologii”, Matematicheskie problemy geofiziki, 6, no. 2, Izd-vo VTs SO AN SSSR, Novosibirsk, 1975, 7–53 | Zbl