Volume 9, no. 2Pages 110 - 116 # On a Model of Oscillations of a Thin Flat Plate with a Variety of Mounts on Opposite Sides

U.A. IskakovaWe consider a model case of stationary vibrations of a thin flat plate, one side of which is embedded, the opposite side is free, and the sides are freely leaned. In mathematical modeling there is a local boundary value problem for the biharmonic equation in a rectangular domain. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Necessary and sufficient conditions of existence of the problem solution are found. Spaces of the ill-posedness of the considered problem are constructed.

Full text- Keywords
- oscillations; thin flat plate; biharmonic equation; boundary value problem; ill-posed problem.
- References
- 1. Kabanikhin S.I. Inverse and Ill-Posed Problems. Siberian Electronic Mathematical Reports, 2010, vol. 7, pp. 380-394.

2. Komkov V. Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems. Berlin, Heidelberg, Springer, 1972.

3. Hadamard J. Le probleme de Cauchy et les equations aux derivees partialles lineaires hyperboliques. Paris, Hermann and Lie, 1932.

4. Lavrent'ev M.M. On the Cauchy Problem for Laplace Equation. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 1956, vol. 20, no. 6, pp. 819-842. (in Russian)

5. Tikhonov A.N. Non-linear Equations of First Kind. Doklady akademii nauk SSSR, 1965, vol. 161, no. 5, pp. 1023-1026. (in Russian)

6. Kal'menov T.Sh., Iskakova U.A. Criterion for the Strong Solvability of the Mixed Cauchy Problem for the Laplace Equation. Doklady Mathematics, 2007, vol. 75, no. 3, pp. 370-373. DOI: 10.1134/S1064562407030118

7. Kal'menov T.Sh., Iskakova U.A. A Method for Solving the Cauchy Problem for the Laplace Equation. Doklady Mathematics, 2008, vol. 78, no. 3, pp. 874-876. DOI: 10.1134/S1064562408060185

8. Kal'menov T.Sh., Iskakova U.A. On a Boundary Value Problem for the Biharmonic Equation. AIP Conference Proceedings, 2015, vol. 1676, no. 020031. DOI: 10.1063/1.4930457.