Abstract
Under consideration is the integro-differential equation of viscoelasticity. The direct problem is to determine the z-component of the displacement vector from the initial boundary value problem for the equation. We assume that the kernel of the integral term of the equation depends on time and a spatial variable x. For determination of the kernel the additional condition is posed on the solution of the direct problem for y = 0. The inverse problem is replaced by an equivalent system of integro-differential equations for the unknown functions. To this system, we apply the method of scales of Banach spaces of analytic functions. The local unique solvability of the inverse problem is proved in the class of functions analytic in x and continuous in t.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. V. Ovsyannikov, “The Nonlinear Cauchy Problem in Scales of Banach Spaces,” Dokl. Akad. Nauk SSSR 200 (4), 789–792 (1971).
L. Nirenberg, Topics in Nonlinear Functional Analysis (Courant Institute Math. Sci., New York, 1974).
V. G. Romanov, “Local Solvability of Some Multidimensional Inverse Problems for Equations of Hyperbolic Type,” Differentsial’nye Uravneniya 25 (2), 275–283 (1989). [Differential Equations 25 (2?), 203–209 (1989)].
V. G. Romanov, “Problem of Determining the Speed of Sound,” Sibir. Mat. Zh. 30 (4), 125–134 (1989)
V. G. Romanov, Siberian Math. J. 30 (4), 598–605 (1989)].
V. G. Romanov, “On the Solvability of Inverse Problems for Hyperbolic Equations in a Class of Functions Analytic in Some of the Variables,” Dokl. Akad. Nauk SSSR 304 (4), 807–811 (1989)
V. G. Romanov, Sov. Math. Dokl. 39 (1), 160–164 (1989)].
D. K. Durdiev, “A Multidimensional Inverse Problem for an Equation with Memory,” Sibir. Mat. Zh. 35 (3), 574–582 (1994)
D. K. Durdiev, Siberian Math. J. 35 (3), 514–521 (1994)].
D. K. Durdiev, “Some Multidimensional Inverse Problems of Memory Determination in Hyperbolic Equations,” Zh. Mat. Fiz. Anal. Geom. 3 (4), 411–423 (2007).
D. K. Durdiev and Zh. Sh. Safarov, “Local Solvability of a Problem of Determination of a Spatial Part of the Multidimensional Kernel of an Integro-Differential Equation of Hyperbolic Type,” Vestnik Samarsk. Gos. Univ. Ser. Fiz.-Mat. Nauki 29 (4), 37–47 (2012).
A. L. Karchevskii and A. G. Fat’yanova, “Numerical Solution of an Inverse Problem for an Elasticity System with Delay for a Vertically Inhomogeneous Medium,” Sibir. Zh. Vychisl. Mat. 4 (3), 259–268 (2001).
V. G. Romanov, “A Two-Dimensional Inverse Problem for a Viscoelasticity Equation,” Sibir. Mat. Zh. 53 (6), 1401–1412 (2012)
V. G. Romanov, Siberian Math. J. 53 (6), 1128–1138 (2012).
D. K. Durdiev and Zh. D. Totieva, “A Problem of Finding a One-Dimensional Kernel of a Viscoelasticity Equation,” Sibir. Zh. Industr. Mat. 16 (2), 72–82 (2013).
R. Courant, Partial Differential Equations (Wiley & Sons, New York, 1962; Mir, Moscow, 1964).
V. G. Romanov, Stability in Inverse Problems (Nauchn. Mir, Moscow, 2005) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2020, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2020, Vol. 23, No. 1, pp. 28–45.
Rights and permissions
About this article
Cite this article
Bozorov, Z.R. The Problem of Determining the Two-Dimensional Kernel of a Viscoelasticity Equation. J. Appl. Ind. Math. 14, 20–36 (2020). https://doi.org/10.1134/S1990478920010044
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478920010044