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The problem of determining a function of the memory of a medium and of the regular part of a pulsed source. (English. Russian original) Zbl 1180.35553
Math. Notes 86, No. 2, 187-195 (2009); translation from Mat. Zametki 86, No. 2, 202-212 (2009).
Summary: Consider the problem of finding two coefficients one of which is under the sign of the integral in the hyperbolic equation and represents the memory of a medium and the other determines the regular part of an impulse source. Additionally, the Fourier transform of the trace of the solution of the direct problem on the hyperplane \(y = 0\) for two different values of the transformation parameter is given. We establish an estimate of the stability of the solution of the inverse problem under consideration and also the uniqueness theorem.
35R30 Inverse problems for PDEs
35R12 Impulsive partial differential equations
35Q74 PDEs in connection with mechanics of deformable solids
74D05 Linear constitutive equations for materials with memory
35A35 Theoretical approximation in context of PDEs
35B35 Stability in context of PDEs
Full Text: DOI
[1] D. K. Durdiev, ”The inverse problem for the three-dimensional wave equation in a medium with memory,” in Mathematical Analysis and Discrete Mathematics (Novosibirsk Gos. Univ., Novosibirsk, 1989), pp. 19–26 [in Russian]. · Zbl 0791.35149
[2] D. K. Durdiev, ”On the question of the well-posedness of the inverse problem for a hyperbolic integro-differential equation,” Sib. Mat. Zh. 33, No.3, 69–77 (1992 Sibirsk. Mat. Zh. 33 (3), 69–77 (1992) [Siberian Math. J. 33 (3), 427–433 (1992)]. · Zbl 0787.45007 · doi:10.1007/BF00970890
[3] A. Lorenzi, ”An identification problem related to a nonlinear hyperbolic integro-differential equation,” Nonlinear Anal. 22(1), 21–44 (1994). · Zbl 0818.93014 · doi:10.1016/0362-546X(94)90003-5
[4] J. Janno and L. Von Wolfersdorf, ”Inverse problems for identification of memory kernels in viscoelasticity,” Math. Methods Appl. Sci. 20(4), 291–314 (1997). · Zbl 0871.35056 · doi:10.1002/(SICI)1099-1476(19970310)20:4<291::AID-MMA860>3.0.CO;2-W
[5] V. G. Romanov, ”On the problem of determining the structure of a layered medium and the form of an impulse source,” Sibirsk. Mat. Zh. 48(4), 867–881 (2007) [SiberianMath. J. 48 (4), 694–706 (2007)]. · Zbl 1164.35528
[6] V. G. Romanov, Inverse Problems of Mathematical Physics (Nauka Moscow, 1984) [in Russian].
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