Search

We consider an inverse spectral problem with the third-order differential equation and the non-separated boundary conditions. Two theorems on the uniqueness of the solution of this problem are proved, and a method for establishing the unknown conditions is obtained, using 19 eigenvalues. The method of approximate calculation of unknown boundary conditions is explained, with the help of an example., Comment: 9 pages

Rods are parts of various devices. If it is impossible to observe the rod directly, the only source of information about possible defects of its fastening can be the natural frequencies of its flexural vibrations. The question arises whether one would be able to detect damage in rod fastening by the natural frequencies of its flexural vibrations. This paper gives and substantiates a positive answer to this question.

The present investigation concerns a disc of varying thickness of whose flexural stiffness $D$ varies with the radius $r$ according to the law $D=D_0 r^m$, where $D_0$ and $m$ are constants. The problem of finding boundary conditions for fastening this disc, which are inaccessible to direct observation, from the natural frequencies of its axisymmetric flexural oscillations is considered. The problem in question belongs to the class of inverse problems and is a completely natural problem of identification of boundary conditions. The search for the unknown conditions for fastening the disc is equivalent to finding the span of the vectors of unknown conditions coefficients. It is shown that this inverse problem is well posed. Two theorems on the uniqueness and a theorem on stability of the solution of this problem are proved, and a method for establishing the unknown conditions for fastening the disc to the walls is indicated. An approximate formula for determining the unknown conditions is obtained using first three natural frequencies. The method of approximate calculation of unknown boundary conditions is explained with the help of three examples of different cases for the fastening the disc (rigid clamping, free support, elastic fixing). Keywords: Boundary conditions, a disc of varying thickness,inverse problem, Plucker condition., Comment: 19 pages

The note contains the proof of the uniqueness theorem for the inverse problem in the case of $n$-th order differential equation., Comment: LaTeX, version 2.09, 3 pages, style article

An uniqueness theorem for the inverse problem in the case of a second-order equation defined on the interval [0,1] when the boundary forms contain combinations of the values of functions at the points 0 and 1 is proved. The auxiliary eigenvalue problems in our theorem are chose in the same manner as in Borg's uniqueness theorem are not as in that of Sadovni\v ci$\check \imath $'s. So number of conditions in our theorem is less than that in Sadovni\v ci$\check\imath$'s., Comment: LaTeX, version 2.09, 5 pages, style article

The differential operators pencil recovery uniqueness theorem is proved in our article. Novelty of this result is not equation coefficient recovery, but boundary conditions coefficients recovery. We show that all conditions of the theorem are essential. We also give some interesting examples., Comment: AmSTeX, version 2.1, 8 pages, amsppt style