Stable methods for an inverse problem in acoustic scattering by an obstacle and an inhomogeneous medium

We consider (in two-dimensional Euclidean space) the scattering of a plane, time-harmonic acoustic wave by an inhomogeneous medium Ω with compact support and a bounded obstacle D lying completely outside of the inhomogeneous medium. We show that one may determine the shape of D and the local speed of sound in Ω from a knowledge of the asymptotic behavior of the scattered wave (i.e. the far field). This is done by considering a constrained optimization problem and employing integral equation and conformal mapping techniques. By assuming a priori that the functions which determine the shape of D and the local speed of sound in Ω lie in given compact sets, we show that the problem is stable, in the sense that the solution of the inverse scattering problem depends continuously on the far field data.     

Optimization of the two-dimensional temperature fields in inhomogeneous bodies with constraints on the phase coordinates

We state a problem of speed-optimal control of the two-dimensional nonstationary temperature regime in inhomogeneous bodies with constraints on the control (the temperature of the heating medium) and the phase coordinates (the temperature of the body, the temperature range, and the heat flux on the surface of the body), and we propose a method for solving the problem numerically. We give as an example the computation of the optimal control of heating of a hollow infinite cylinder under constraints on the temperature range. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 119–123.     

Interpolation for solutions of the Helmholtz equation

We study the interpolation problem for solutions of the two-dimensional Helmholtz equation, which are sampled along a line. The data are the function values and the normal derivatives at a discrete set of point sensors. A wave transform is used, analogous to the common Fourier transform. The inverse wave transform defines the Hilbert space for oscillatory Helmholtz solutions. We thereby introduce an interpolant that has some advantages over the usual sinc x in the Whittaker–Shannon sampling in one dimension; in particular, coefficients of the two-dimensional solution are invariant under translations and rotations of the sampling line. The analysis is relevant for the optical sampling problem by sensors on a screen. © 1995 John Wiley & Sons, Inc.     

Integral Equation Method for Modeling of Two-Dimensional Geoelectricity Problems

The article computes the electromagnetic field on the surface of a layered medium with a local nonhomogeneity. The problem is transformed from three- to two-dimensional and the singulari-ties are investigated using the integral equation method. The proposed algorithm efficiently sim-ulates two-dimensional H-polarization fields by solving a system of integral equations. The method is particularly effective for solving inverse problems. __________ Translated from Prikladnaya Matematika i Informatika, No. 18, pp. 5–16, 2004.     

Numerical study of the two-dimensional problem of the theory of elasticity using the boundary-element method

We propose a method of solving the two-dimensional problem of the theory of elasticity by a direct boundary-element method. We apply Galerkin's method with linear and quadratic approximations of the forces and displacements. We give the numerical results of solving a model problem that shows the effectiveness of the proposed approach.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 60–63.     

A strongly degenerate system involving an equation of parabolic type and an equation of elliptic type

Abstract

We show the existence of weak solutions in an elliptic region in the self-similar plane to the two-dimensional Riemann problem for the pressure-gradient system of the compressible Euler system. The two-dimensional Riemann problem we study is the interaction of two forward rarefaction waves, which are adjacent to a common vacuum that occupies a sectorial domain of 90 degrees. We assume the origin is on the boundary of the domain. In addition, the domain is open, bounded, and simply connected with a piecewise C ^2,α boundary. We resolve the difficulty that arises from the fact that the origin is on the boundary of the domain.     

Inverse problem for the system of thermoelasticity equations for a vertically inhomogeneous cohesionless medium with memory

We consider the problem of finding the integrand h(t), t ∈ [0, T], occurring in the system of thermoelasticity equations for a cohesionless medium with memory. The medium density and the Lamé parameters are assumed to be functions of one variable. We prove the theorem on the local unique solvability of the problem and obtain an estimate for the conditional stability of the solution.     

Solution of the two-dimensional integrodifferential equation of the heat conduction problem for an unbounded body with a thin thermally conducting inclusion

We determine the temperature field in an unbounded body with a thin heat-conducting foreign inclusion. We propose a method of solution of the corresponding two-dimensional singular integrodifferenlial equation of the problem based on the study of the behavior of the solution in a neighborhood of the boundary of the inclusion.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 78–83.     

Determination of the coefficient of heat output in the inverse problem for the heat equation

To determine the variable coefficient of heat output in a boundary condition of third kind we propose applying a representation of the solution of the one-sided heat conduction problem obtained by the symbolic operator method. We consider the cases of the one- and two-dimensional inverse problems.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 9–13.     

On the time optimization problem for two-dimensional control systems

We consider a time optimization problem for two-dimensional nonlinear control systems. For a constraint for the control vector u we take a convex polygon U. We obtain effective sufficient conditions under which the time-optimal control is equivalent in the Lebesgue sense to a piecewise constant control that has finitely many points of discontinuity and ranges in the set of vertices of U. We give examples illustrating the results.     

Regularity results and optimal control problems for the perturbation of boussinesq equations of the ocean

We study in this article a method which computes the variability of current, density and pressure in an oceanic domain. The equations are of Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the density. They are linearized around a given mean circulation and modified by the Boussinesq approximation: density variations are neglected except in the terms of gravity acceleration. The existence and uniqueness of a solution are proved for two sets of equations: first the three-dimensional problem and then the two-dimensional cyclic problem derived by assuming a sinusoidal x-dependence for the perturbation of mean flow. The latter corresponds to a modelization of tropical instability waves which are illustrated by the El Nino phenomenon.

The value of the pressure p on the surface of ocean is of great interest for physical interpretation. To define that quantity, it is necessary to have the regularity p ? H ¹. We have proved that the perturbation (u,ρ,p) of mean circulation is such that: u ? L ²(0T,H ²), ρ ? L ²(0,T H ²) and p ? L ² L ²(0,T H ¹), provided the perturbation of the windstress is sufficiently regular and satisfies compatibility relations. It is proved by means of an extension method, with even-odd reflection. We then develop a problem of control. The observation is the Variability of pressure on the surface of ocean. The control is the variability of windstress f, which acts as to forcing of the perturbation. We prove the existence and uniqueness of an optimal control, which is characterized by a set of equations including the direct problem and the adjoint problem. These results are valid for the three-dimensional problem and the two-dimensional cyclic problem.     

On certain solutions of the diffusion equation for a two-dimensional Brownian particle density

We find approximate solutions of the diffusion equation for a two-dimensional Brownian particle density. We obtain expressions for the corrections connected with the fluctuations of the transfer coefficients, which can be applied in particular to describe the fluctuations of the concentration of Brownian particles in small volumes of the medium. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Entropy solutions to the Verigin ultraparabolic problem

We study the Cauchy problem for the two-dimensional ultraparabolic model of filtration of a viscous incompressible fluid containing an admixture, with diffusion of the admixture in a porous medium taken into account. The porous medium consists of the fibers directed along some vector field n ^⊥. We prove that if the nonlinearity in the equations of the model and the geometric structure of fibers satisfy some additional “genuine nonlinearity” condition then the Cauchy problem with bounded initial data has at least one entropy solution and the fast oscillating regimes possible in the initial data are promptly suppressed in the entropy solutions. The proofs base on the introduction and systematic study of the kinetic equation associated with the problem as well as on application of the modification of Tartar H-measures which was proposed by Panov.     

Vorticity transport in a viscoelastic fluid in the presence of suspended particles through porous media

We consider the transport of vorticity in an Oldroydian viscoelastic fluid in the presence of suspended magnetic particles through porous media. We obtain the equations governing such a transport of vorticity from the equations of magnetic fluid flow. It follows from these equations that the transport of solid vorticity is coupled to the transport of fluid vorticity in a porous medium. Further, we find that because of a thermokinetic process, fluid vorticity can exist in the absence of solid vorticity in a porous medium, but when fluid vorticity is zero, then solid vorticity is necessarily zero. We also study a two-dimensional case.     

On the Galerkin-Method in Intermediate Spaces for the Initial Boundary-Value Problem of the Navier-Stokes Equations in two Dimensions

We consider a GALERKIN scheme for the two-dimensional initial boundary-value problem (P) of the NAVIER-STOKES equations, derive a priori-estimates for the approximations in interpolation spaces between “standard spaces” as occuring in the theory of weak solutions and obtain well-posedness of (P) with respect to the interpolation spaces. As a by-product, we obtain various estimates for the solutions in a relatively simple way. The main tool is interpolation of non-linear operators.     

Existence of a stationary symmetric solution of the two-dimensional Navier-Stokes equations with a given head and with nonzero fluxes

We consider a stationary boundary value problem for the Navier-Stokes equations of a homogeneous incompressible fluid in a two-dimensional bounded domain with boundary consisting of connected components Γ _i . On each part Γ _i , we specify the tangent component of the flow velocity vector, the total flow head (up to an additive constant), and the fluid flux through Γ _i . For the case in which the domain and the original data are symmetric around some line, we prove the existence of a solution of the problem with such a symmetry. We also present some results on the solvability in the nonsymmetric case.     

A study of the bending of a short beam

We consider the two-dimensional problem of static deformation ofa short beam under the action of a self-balanced load. We propose an approximate method of solution based on a variational approach and a special choice of the stress function. We prove that the resulting boundary-value problem for a system of ordinary differential equations is well-posed. For special cases of the boundary conditions we give an analysis of the solutions. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 65–73.     

Self-similar problem of Cauchy-Poisson waves at an inclined shore : PMM vol. 37, n6, 1973, pp. 1035–1039

We consider a two-dimensional problem concerning Cauchy-Poisson waves at an inclined shore in the case of an initial disturbance concentrated near the shore edge. We study the behavior of the solution near the shore and at large distances from it.Numerous investigations, devoted to the study of standing and progressive waves on an inclined shore, are described in [1]. A two-dimensional problem concerning nonstationary waves on a shore with an angle of inclination γ = π/2n, where n is an integer, was analyzed in [2, 3]. We consider below a case in which the angle of inclination is commensurable with λ/2, subject to the condition that the initial disturbance is concentrated in the vicinity of the shore edge, so that the problem may be considered self-similar.     

On potential theory for two-dimensional quasistatic problems of uncoupled thermoelasticity

The construction of potential theory for two-dimensional quasistatic problems of uncoupled thermoelasticity is carried out by considering the full system of differential equations of the problem as a nonselfadjoint differential operator. Green's second formula for this operator is interpreted as a duality theorem that differs from Mizel's duality theorem. In the case of a homogeneous isotropic medium we construct new integral equations for the basic initial-boundary value problems.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 48–52.