Abstract
We consider the problem of finding the kernel K(t), for t ∈ [0, T], in the integrodifferential system of electroviscoelasticity. We assume that the coefficients depend only on one spatial variable. Replacing the inverse problem with an equivalent system of integral equations, we apply the contraction mapping principle in the space of continuous functions with weighted norms. We prove a global unique solvability theorem and obtain a stability estimate for the solution to the inverse problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dieulesaint E. and Royer D., Elastic Waves in Solids [Russian], Nauka, Moscow (1982).
Yakhno V. G. and Merazhov I. Z., “Direct problems and a one-dimensional inverse problem of electroelasticity for ‘slow’ waves,” Siberian Adv. Math., vol. 10, no. 1, 87–150 (2000).
Romanov V. G., “Stability estimates for the solution to the problem of determining the kernel of a viscoelastic equation,” J. Appl. Ind. Math., vol. 6, no. 3, 360–370 (2012).
Lorensi A. and Romanov V. G., “Stability estimates for an inverse problem related to viscoelastic media,” J. Inv. Ill-Posed Probl., vol. 14, no. 1, 57–82 (2006).
Durdiev D. K. and Totieva Zh. D., “The problem of determining the multidimensional kernel of the viscoelasticity equation,” Vladikavkaz. Mat. Zh., vol. 17, no. 4, 18–43 (2015).
Durdiev D. K. and Safarov Zh. Sh., “Inverse problem of determining the one-dimensional kernel of the viscoelasticity equation in a bounded domain,” Math. Notes, vol. 97, no. 6, 867–877 (2015).
Durdiev D. K. and Totieva Zh. D., “The problem of determining the one-dimensional kernel of the viscoelasticity equation,” Sib. Zh. Ind. Mat., vol. 16, no. 2, 72–82 (2013).
Janno J. and von Wolfersdorf L., “An inverse problem for identification of a time- and space-dependent memory kernel in viscoelasticity,” Inverse Probl., vol. 17, no. 1, 13–24 (2001).
Colombo F. and Guidetti D., “Some results on the identification of memory kernels,” Oper. Theory: Adv. Appl., vol. 216, 121–138 (2011).
Durdiev D. K., “An inverse problem for determining two coefficients in an integrodifferential wave equation,” Sib. Zh. Ind. Mat., vol. 12, no. 3, 28–49 (2009).
Tuaeva Zh. D., “The many-dimensional mathematical seismic model with memory,” in: Studies on Differential Equations and Mathematical Modeling [Russian], VNTs RAN, Vladikavkaz, 2008, 297–306.
Yakhno V. G., Inverse Problems for Differential Elasticity Equations [Russian], Nauka, Novosibirsk (1990).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2017 Durdiev D.K. and Totieva Zh.D.
Rights and permissions
About this article
Cite this article
Durdiev, D.K., Totieva, Z.D. The problem of determining the one-dimensional kernel of the electroviscoelasticity equation. Sib Math J 58, 427–444 (2017). https://doi.org/10.1134/S0037446617030077
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446617030077