We give a complete description of well-posed solvable boundary-value problems for the Laplace operator in the disk and in the punctured disk. We present formulas for resolvents of wellposed problems for the Laplace operator in the disk.

In this paper, the models of Euler–Bernoulli non-uniform beams with the axial loads on the Winkler foundations are considered. The non-uniform beam in the model is described by three variable parameters/coefficients: bending stiffness, foundation and beam mass per unit length. The key finding of this study is the clear demonstration of how the agreed symmetry of variable parameters affects the spectral properties of a problem. The qualitative results for the symmetric equivalence (factorisation of sets of eigenvalues and eigenfunctions) of eigenvalues of non-uniform beams for two types of fixing at the ends (clamped-clamped and hinged-hinged) have been obtained. In order to demonstrate equivalence, a hybrid algorithm has been devised, based on the qualitative spectral properties of fourth-order ordinary differential equations and axial load calculations. The results have been validated using examples on the Maple computer package and compared with the experimental measurements.

In this paper, the models of Euler–Bernoulli beams on the Winkler foundations are considered. The novelty of the research is in consideration of the models with an arbitrary variable coefficient of foundation. Qualitative results that influence the symmetry of the coefficient of foundation on the spectral properties of the corresponding problems are obtained, for which specific variable coefficients of foundation are tested using numerical calculations. Three types of fixing at the ends are studied: clamped-clamped, hinged-hinged and free-free. The conditions of the stiffness and types of beam fixing have been found for the set of eigenvalues of boundary value problems on a full segment and can be represented as two groups of the eigenvalues of certain problems on a half segment. Such qualitative spectral properties of a mechanical system can contribute to the creation of various algorithms for nondestructive testing, which are widely used in technical acoustics.

The inverse problem of determining the weight of three intermediate masses on a uniform beam from the known three natural frequencies has been solved. The performed numerical analysis allows restoring the value of only the second mass in a unique way. The inverse problem of determining the weight of three intermediate masses is solved uniquely except in the case when the first and the third masses are located geometrically symmetric relative to the middle of the beam. The hybrid algorithm for the unique solving inverse problem of determining the weight of three intermediate masses has been developed. The first three natural frequencies of the beam are calculated numerically by using the Maple computer package. Analytical relations between the masses are found.