Geodesic
The problem of determining the one-dimensional kernel of the viscoelasticity equation
Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 2, pp. 72-82.

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We consider the integrodifferential system of viscoelasticity equations. The direct problem consists in determining the displacement vector from the initial boundary value problem for this system. Under the assumption that the coefficients of the equation depend only on one space variable $x_3$, the system is reduced to an equation for one component $u_1(x_3,t)$. For this equation, we investigate the problem of finding the kernel belonging to the integral part of the equation. For its determination, an additional condition is given on $u_1(x_3,t)$ for $x_3=0$. The inverse problem is replaced by an equivalent system of integral equations for unknown functions. To this system, we apply the contraction mapping principle. A theorem of global unique solvability is proved and a stability estimate of a solution to the inverse problem is obtained.
Keywords: inverse problem, stability, delta-function
Mots-clés : Lamé coefficients, kernel. 
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D. K. Durdiev; Zh. D. Totieva. The problem of determining the one-dimensional kernel of the viscoelasticity equation. Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 2, pp. 72-82. http://geodesic.mathdoc.fr/item/SJIM_2013_16_2_a6/

[1] Janno J., von Wolfersdorf L., “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

[2] Romanov V. G., “Otsenka ustoichivosti resheniya v zadache ob opredelenii yadra uravneniya vyazkouprugosti”, Sib. zhurn. industr. matematiki, 15:1 (2012), 86–98

[3] Durdiev D. K., “Obratnaya zadacha dlya sistemy uravnenii termouprugosti v vertikalno-neodnorodnoi nesvyaznoi srede s pamyatyu”, Differents. uravneniya, 45:9 (2009), 1229–1236 | MR | Zbl

[4] Tuaeva Zh. D., “Mnogomernaya matematicheskaya model seismiki s pamyatyu”, Issledovaniya po differents. uravneniyam i mat. modelirovaniyu, izd. VNTs RAN, Vladikavkaz, 2008, 297–306

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

[6] Yakhno V. G., Obratnye zadachi dlya differentsialnykh uravnenii uprugosti, Nauka, Novosibirsk, 1990 | MR | Zbl