卷 214, 编号 3 (2016)

The paper has a scientific-methodical nature. The classical soap film shape (minimal surface) problem is considered, the film being stretched between two parallel coaxial rings. An analytic approach based on the relations to the Sturm–Liouville problem is proposed. An interpretation of the energy terms of the classical Goldschmidt condition is discussed. The appearance of a soliton potential when analyzing the second variation is noticed. Bibliography: 3 titles.

The confluent Heun equation with an added apparent singular point is under consideration. A new integral transform connecting solutions of this equation with different parameters is obtained. The kernel of this transform is a suitable solution of the confluent hypergeometric equation. Bibliography: 22 titles.

Using the dimension reduction procedure, a one-dimensional model of a periodic blood flow in the artery through a small hole in a thin elastic wall to a spindle-shaped hematoma, is constructed. This model is described by a system of two parabolic and one hyperbolic equations provided with mixed boundary and periodicity conditions. The blood exchange between the artery and the hematoma is expressed by the Kirchhoff transmission conditions. Despite the simplicity, the constructed model allows us to describe the damping of a pulsating blood flow by the hematoma and to determine the condition of its growth. In medicine, the biological object considered is called a false aneurysm. Bibliography: 15 titles.

The acoustic problem of diffraction by two wedges with different wave velocities is studied. It is assumed that the wedges with parallel edges have a common part of the boundary and the second wedge is shifted with respect of the first one in the orthogonal to the edges direction along the common part of the boundary. The wave field is governed by the Helmholtz equations. On the polygonal boundary, separating these shifted wedges from the exterior, the Dirichlet boundary condition is satisfied. The wave field is excited by an infinite filamentary source, which is parallel to the edges. In these conditions, the problem is effectively two-dimensional. The Kontorovich–Lebedev transform is applied to separate the radial and angular variables and to reduce the problem at hand to integral equations of the second kind for so-called spectral functions. The kernel of the integral equations given in the form of an integral of the product of Macdonald functions is analytically transformed to a simplified expression. For the problem at hand, some reductions of the equations are also discussed for the limiting or degenerate values of parameters. Making use of an alternative integral representation of the Sommerfeld type, expressions for the scattering diagram are then given in terms of spectral functions. Bibliography: 24 titles.

The evolution of the dynamical system under consideration is governed by the wave equation ρu_tt − (γu_x)_x + Au_x + B_u = 0, x>0, t > 0, with the zero initial Cauchy data and Dirichlet boundary control at x = 0. Here, ρ, γ, A, B are smooth 2 × 2–matrix-valued functions of x; ρ = diag {ρ₁, ρ₂} and γ = diag {γ₁, γ₂} are matrices with positive entries; u = u(x, t) is a solution (an ℝ²-valued function). In applications, the system corresponds to one-dimensional models, in which there are two types of wave modes, which propagate with different velocities and interact with each other. The “input→state” correspondence is realized by the response operator R : u(0, t) _→ γ(0)u_x(0, t), t ≥ 0, which plays the role of inverse data. The representations for the coefficients A and B, which are used for their determination via the response operator, are derived. An example of two systems with the same response operator is given, where in the first system the wave modes do not interact, whereas in the second one the interaction does occur. Bibliography: 3 titles.

New relations are found between the spheroidal and spherical wave functions, as well as between the spheroidal functions related to different spheroidal coordinate systems. The systems must have a common origin of coordinates and a common symmetry axis of coordinate surfaces. The applicability ranges of the relations obtained are discussed. Numerical test calculations have demonstrated a high efficiency of the relations, specifically those for wave functions including the radial functions of the first kind. As a particular case, relations are considered between prolate and oblate spheroidal wave functions including radial functions of the first and second kinds. These relations are necessary for solving the light scattering problem for nonconfocal layered spheroidal particles. Bibliography: 15 titles.