Том 55, № 9 (2019)

We consider the problem about leaky waves in an inhomogeneous waveguide structure. This problem is reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. A variational problem statement is used to determine the solution. The variational problem is reduced to studying an operator function. The properties of this operator function necessary for analyzing its spectral characteristics are investigated. Theorems about the discreteness of the spectrum and the distribution of the characteristic numbers of the operator function on the complex plane are proved. The existence of infinitely many damped leaky waves in a cylindrical waveguide is established.

For a quasilinear hyperbolic equation containing a numerical parameter, we consider the inverse problem of determining an unknown function occurring in the free term of the equation. The inverse problem is reduced to a system of integral equations for the unknown function and the solution of the hyperbolic equation. We prove the existence and uniqueness of the solution of this system and of the inverse problem. The asymptotics of the solution of the system is studied for small values of the parameter occurring in the hyperbolic equation.

Abstrac

We study a nonlinear integral equation arising from the parametric closure for the third spatial moment in the Dieckmann-Law model of stationary biological communities. The existence of a fixed point of the integral operator defined by this equation is analyzed. The noncompactness of the resulting operator is proved. Conditions are stated under which the equation in question has a nontrivial solution.

Integral representations for the velocity and pressure fields are constructed in the problem on a three-dimensional filtration flow of a viscous fluid obeying the Darcy-Brinkman law in a piecewise homogeneous medium. This problem is reduced to a system of boundary integral equations.

This work continues the research devoted to considering interval Taylor models (TM) as applied to proving the existence of periodic trajectories in systems of ordinary differential equations (ODEs). The Taylor models are used here within a topological approach to constructing an isolating set for the ODE phase flow. We prove auxiliary assertions and propose constructive algorithms for validated numerics over TMs with the aim to expand the domain of their applicability. Necessary topological assertions are proved in order to establish the properties of isolating sets constructed with the aid of TMs. As a result, constructive algorithms are formulated and the main theorem is proved. This theorem makes it possible to construct and verify the homotopy equivalence of the isolating set and the one-dimensional sphere and the homotopy of the mapping of the isolating set into itself to the identity mapping for given systems of ODEs. We also prove the computational complexity of the main algorithm and provide an example of its usage.

A quadrature formula for the simple layer potential with smooth density specified on a closed or open surface is derived. The formula uniformly approximates the potential near the surface and inherits its continuity as the observation point tends to the surface from the interior of the domain; this has been confirmed by numerical tests. The proposed quadrature formula is much more accurate in calculating the potential near the surface than the standard quadrature formulas, a fact that has also been confirmed by numerical tests.

We consider numerical methods for 3D singular integral equations with time delay, which describe a broad class of problems of interaction of a nonstationary electromagnetic field with a bounded material medium. An efficient solution method for linear dispersionless media is proposed.

We obtain a criterion for the boundedness and compactness of a class of sets in the space L[0,∞).