Abstract: The integro-differential system of viscoelasticity equations is considered. The direct problem of determining of the displacements vector from the initial-boundary problem for this system is formulated. It is assumed that the kernel in the integral part depends on both the time and the space variable \(x_2\). For its determination an additional condition relative to the first component of the displacements vector with \(x_3=0\) is posed. The inverse problem is replaced by the equivalent system of integral equations. The study is based on the method of scales of Banach spaces of analytic functions. The theorem on local unique solvability of the inverse problem is proved in the class of functions analytic on the variable \(x_2\) and continuous on \(t\).
For citation: Totieva Zh. D., Durdiev D. Q. The problem of determining the multidimensional kernel of viscoelasticity equation. Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol.
17, no. 4, pp.18-43.
DOI 10.23671/VNC.2015.4.5969
1. Tuaeva Zh. D. Mnogomernaja matematicheskaja model' sejsmiki s
pamjat'ju. Mat. Forum. T. 1, Ch. 2. Issledovanija po Dif.
Uravnenijam i Mat. Modelirovaniju [Math. Forum. Vol. 1, part 1.
Studies on Dif. Equations and Math. Modeling], Vladikavkaz, VNC RAN,
2008, pp. 297-306 (Russian).
2. Durdiev D. K., Totieva Zh. D. The problem of determining the
one-dimensional kernel of the viscoelasticity equation. Sib. Zh.
Ind. Mat. [J. Applied and Industrial Math.], 2013, vol. 16, no. 2,
pp. 72-82 (Russian).
3. Ovsjannikov L. V. Singuljarnyj operator v shkale banahovyh
prostranstv. Dokl. AN SSSR [Dokl. Math.], 1965, vol. 163, no. 4,
pp. 819-822 (Russian).
4. Ovsjannikov L. V. Nelinejnaja zadacha Koshi v shkalah banahovyh
prostranstv. Dokl. AN SSSR [Dokl. Math.], 1971, vol. 200, no. 4, pp.
789-792 (Russian).
5. Nirenberg L. Topics in Nonlinear Functional Analysis, N. Y.,
Courant Institute Math. Sci., New York Univ., 1974, 259 p.
6. Romanov V. G. Local solvability of some multidimensional inverse
problems for equations of hyperbolic type. Dif. Equations [Dif.
Uravneniya, 1989, vol. 25, no. 2, pp. 275-284], 1989, vol. 25, no.
2, pp. 203-205.
7. Romanov V. G. Problem of determination the speed of sound.
Siberian Math. J. [Sib. Mat. J., 1989, vol. 30, no. 4, pp. 125-134],
1989, vol. 30, no. 4, pp. 598-605.
8. Romanov V. G. On the solvability of inverse problems for
hyperbolic equations in a class of functions analytic in some of
variables. Soviet Math. Dokl. [Dokl. AN SSSR, 1989, vol. 304, no. 4,
pp. 807-811], 1989, vol. 39, no. 1, pp. 160-164.
9. Durdiev D. K. A multidimensional inverse problem for an equation
with memory. Siberian Math. J. [Sib. Mat. J., 1994, vol. 35, no. 3,
pp. 574-582], 1994, vol. 35, no. 3, pp. 514-521.
10. Durdiev D. K. Some multidimensional inverse problems of memory
determination in hyperbolic equations. Zh. Mat. Fiz. Anal. Geom.,
2007, vol. 3, no 4, pp. 411-423.
11. Durdiev D. K., Safarovb Zh. Sh. The local solvability of a
problem of determining the spatial part of a multidimensional kernel
in the integro-differential equation of hyperbolic type. Vestn.
Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State
Technical Univ. Ser. Phys. and Math. Sci.], 2012, no. 4(29), pp.
37-47 (Russian).