Open Access Open Access  Restricted Access Access granted  Restricted Access Subscription Access

Vol 55, No 9 (2019)

Partial Differential Equations

Existence of an Infinite Spectrum of Damped Leaky TE Waves in an Open Inhomogeneous Cylindrical Metal–Dielectric Waveguide

Smirnov Y.G., Smol’kin E.Y.

Abstract

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.

Differential Equations. 2019;55(9):1125-1133
pages 1125-1133 views

Generalization of the Robin Problem for the Laplace Equation

Turmetov B.K.

Abstract

The solvability of a new class of boundary value problems for the Laplace equation is studied. The problem considered is a generalization of the classical Robin problem. Exact conditions are established for the solvability of the problem, and integral solution representations are constructed for various cases of data.

Differential Equations. 2019;55(9):1134-1142
pages 1134-1142 views

Integral and Integro-Differential Equations

System of Integral Equations for Solving an Inverse Problem for a Quasilinear Hyperbolic Equation

Denisov A.M.

Abstract

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.

Differential Equations. 2019;55(9):1143-1149
pages 1143-1149 views

Asymptotics of Solutions in the Problem about Small Motions of a Compressible Maxwell Fluid

Zakora D.A.

Abstract

A model of a viscoelastic compressible Maxwell fluid is studied. This model is described by a system of partial integro-differential equations with appropriate boundary and initial conditions. An abstract analog of the problem under study is also considered. It is proved that a uniformly exponentially stable C0-semigroup emerges in this problem. Based on this fact, an estimate for the solution of the evolution problem is derived in the case where the external load is close to being almost periodic.

Differential Equations. 2019;55(9):1150-1163
pages 1150-1163 views

Application of the Leray-Schauder Principle to the Analysis of a Nonlinear Integral Equation

Nikolaev M.V., Nikitin A.A.

Abstract

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.

Differential Equations. 2019;55(9):1164-1173
pages 1164-1173 views

Generalized Transmission Problem for Two-Dimensional Filtration Flows in an Anisotropic Inhomogeneous Layer

Piven’ V.F.

Abstract

We state and study a transmission boundary value problem for two-dimensional filtration flows in a piecewise anisotropic inhomogeneous layer of a porous medium. The layer is characterized by a generally nonsymmetric conductivity (permeability) tensor with components that undergo discontinuity on some smooth curve (the transmission line). The tensor components are modeled by a function of the coordinates that undergoes a discontinuity on the transmission line but is continuously differentiable outside of it. We consider a layer with separated anisotropy and inhomogeneity. Using a nonsingular affine transformation of the coordinates, we state the problem for a complex potential in canonical form, which considerably simplifies the analysis of the problem. The sources–sinks of the flow are set arbitrarily; they do not lie on the transmission line and are modeled by the singular points of the complex potential. The problem is reduced to a system of two singular integral equations if the discontinuity in the layer conductivity along the transmission line is variable and to one singular integral equation if the discontinuity is constant. The problem is of practical interest, for example, in extracting water (or oil) from natural piecewise anisotropic inhomogeneous layers (strata) of soil.

Differential Equations. 2019;55(9):1174-1181
pages 1174-1181 views

Methods of Potential Theory in a Filtration Problem for a Viscous Fluid

Setukha A.V., Tretyakova R.M., Bocharov G.A.

Abstract

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.

Differential Equations. 2019;55(9):1182-1197
pages 1182-1197 views

Numerical Methods

Algorithms for Constructing Isolating Sets of Phase Flows and Computer-Assisted Proofs with the Use of Interval Taylor Models

Evstigneev N.M., Ryabkov O.I.

Abstract

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.

Differential Equations. 2019;55(9):1198-1217
pages 1198-1217 views

Special Version of the Spline Method for Integral Equations of the Third Kind-

Gabbasov N.S.

Abstract

We study a linear integral equation of the third kind with a coefficient having power-order zeros. A special generalized version of the spline method is proposed for approximately solving this equation in a generalized function space. We show that the method is accuracy order optimal.

Differential Equations. 2019;55(9):1218-1225
pages 1218-1225 views

Quadrature Formula for the Simple Layer Potential

Krutitskii P.A., Fedotova A.D., Kolybasova V.V.

Abstract

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.

Differential Equations. 2019;55(9):1226-1241
pages 1226-1241 views

Numerical Method for Some Singular Integro-Differential Equations

Rasol’ko G.A., Sheshko S.M., Sheshko M.A.

Abstract

Numerical solution schemes are constructed and justified for two singular integro-differential equations containing an integral, understood in the sense of the Cauchy principal value, over an interval of the real axis. An integral equation with logarithmic kernel of a special form is studied and approximately solved. Uniform error estimates for the approximate solutions are obtained.

Differential Equations. 2019;55(9):1242-1249
pages 1242-1249 views

Numerical Methods for a Nonstationary 3D Singular Integral Equation of Electrodynamics

Samokhin A.B., Samokhina A.S., Kobayashi K.

Abstract

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.

Differential Equations. 2019;55(9):1250-1257
pages 1250-1257 views

Short Communications

Criterion for the Boundedness and Compactness of a Class of Sets in L[0,∞)

Otelbaev M., Sultanaev Y.T., Zhusupova D.S.

Abstract

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

Differential Equations. 2019;55(9):1258-1261
pages 1258-1261 views