Limiting Amplitude principle in the scattering by wedges

We consider a nonstationary scattering of plane waves by a wedge. We prove that the Sommerfeld‐type integral, constructed in (Math. Meth. Appl. Sci. 2005; 28 :147–183; Proc. Int. Seminar ‘Day on Diffraction‐2003’, University of St. Petersburg, 2003; 151–162), is a classical smooth solution from a functional space, and prove the Limiting Amplitude principle. Copyright © 2006 John Wiley & Sons, Ltd.     

Axially Symmetric Boundary Value Problem Describing the Distribution of Charges in Semiconductors

We prove that bifurcation solutions to the stationary problem exist and can be extended with respect to parameter. The problem under consideration has a solution with the so-called interior transition layer phenomenon. In the nonstationary case, we establish the existence and uniqueness of a solution for any t > 0. Under certain assumptions, the nonstationary problem determines a dynamical system in some compact set. Bibliography: 9 titles.     

Solvability of the initial-boundary value problem for an integrodifferential equation

Under study is the well-posedness of the Cauchy problem for the nonstationary radiation transfer equation with generalized matching conditions at the interface between the media. We prove the existence of a unique strongly continuous semigroup of resolvents, estimate its order of growth, and consider the question of stabilization of the nonstationary solution.     

Nonstationary poiseuille-type solutions for the second-grade fluid flow

We study solutions to the nonstationary equations describing the motion of the second-grade fluid in a twodimensional infinite channel and in a three-dimensional rotationally symmetric pipe. We prove the existence of a unique Poiseuille-type solution having a given flow rate (flux) and investigate the behavior of this solution as t → ∞.     

Existence and Regularity of Purely Viscous Polymer Flow

In this article we study purely viscous nonstationary flow of quasi-Newtonian fluids with share dependent viscosity obeying the power-law and the Carreau's law. We prove the existence and regularity of the solution using the Galerkin's procedure.     

On a Strong Solution in the Method of Spherical Harmonics for a Nonstationary Transport Equation

We prove existence of a strong generalized solution in the Sobolev space W ₂ ¹ to the nonstationary problem for the system of the method of spherical harmonics (MSH) corresponding to the radiation transport problem.     

Stefan problem for the laplace equation with regard for the curvature of the free boundary

We consider a nonstationary problem with free boundary for an elliptic equation in the case where the value of the required function on an unknown boundary is proportional to the curvature of this boundary. We prove the existence of a solution in the small with respect to time in the spaces of smooth functions.     

Aspects of electromagnetic scattering in chiral media

In this work, we study two operators that arise in electromagnetic scattering in chiral media. We first consider electromagnetic scattering by a chiral dielectric with a perfectly conducting core. We define a chiral Calderon‐type surface operator in order to solve the direct electromagnetic scattering problem. For this operator, we state coercivity and prove compactness properties. In order to prove existence and uniqueness of the problem, we define some other operators that are also related to the chiral Calderon‐type operator, and we state some of their properties that they and their linear combinations satisfy. Then we sketch how to use these operators in order to prove the existence of the solution of the direct scattering problem. Furthermore, we focus on the electromagnetic scattering problem by a perfect conductor in a chiral environment. For this problem, we study the chiral far‐field operator that is defined on a unit sphere and contains the far‐field data, and we state and prove some of its properties that are preliminaries properties for solving the inverse scattering problem. Copyright © 2016 John Wiley & Sons, Ltd.     

The Impedance Boundary Value Problem for the Helmholtz Equation in a Half-Plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L_∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5]. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Integral equation methods in electromagnetic scattering from anisotropic media

We investigate scattering of time‐harmonic electromagnetic waves by an anisotropic inhomogeneous medium. The problem is equivalently transformed into a system of strongly singular integral equations. The uniqueness and existence of a solution is shown and we examine the regularity of the solution by means of integral equations. We also prove the analyticity of the scattered field with respect to the refractive matrix and give a characterization of the derivatives in terms of solutions to anisotropic scattering problems. Copyright © 2000 John Wiley & Sons, Ltd.     

Sommerfeld's solution as the limiting amplitude and asymptotics for narrow wedges

We analyze the Sommerfeld solution to the stationary diffraction by a half‐plane. We prove that this solution is the limiting amplitude for time‐dependent scattering of incident plane waves with a broad class of the profile functions. We also show that this solution is the asymptotics of the limiting amplitudes of solutions to time‐dependent scattering problem with narrow wedges when the angle of the wedge tends to zero.     

Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids

This paper analyzes an initial/boundary value problem for a system of equations modelling the nonstationary flow of a nonhomogeneous incompressible asymmetric (polar) fluid. Under conditions similar to those usually imposed to the nonhomogeneous 3D Navier–Stokes equations, using a spectral semi-Galerkin method, we prove the existence of a local in time strong solution. We also prove the uniqueness of the strong solution and some global existence results. Several estimates for the solutions and their approximations are given. These can be used to find useful error bounds of the Galerkin approximations.     

Bell-shaped nonstationary refinable ripplets

We study the approximation properties of the class of nonstationary refinable ripplets introduced in Gori and Pitolli (2008). These functions are the solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions, convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the C ² nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented.     

Inverse scattering problem for a wave equation with absorption

We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1580–1584, November, 2007.     

On a parabolic–elliptic chemotactic model with coupled boundary conditions

This paper deals with a nonlinear system of parabolic–elliptic type with a logistic source term and coupled boundary conditions related to pattern formation. We prove the existence of a unique positive global in time classical solution. We also analyze the associated stationary problem. Moreover it is proved, under the assumption of sufficiently strong logistic damping, that there is only one nonzero homogeneous equilibrium, and all the solutions to the nonstationary problem tend to this steady state for large times.     

On the solution of a Schrödinger type equation by a projection-difference method with an implicit Euler scheme with respect to time

A linear nonstationary Schrödinger type problem in a separable Hilbert space is approximately solved by a projection-difference method. The problem is discretized in space by the Galerkin method using finite-dimensional subspaces of finite-element type, and an implicit Euler scheme is used with respect to time. We establish error estimates uniform with respect to the time grid for the approximate solutions; as to the spatial variables, the estimates are given in the norm of the original space as well as in the energy norm. The estimates considered here not only permit one to prove the convergence of approximate solutions to the exact solution but also give a numerical characterization of the convergence rate.     

Joint estimates of quasi-integrals of three-dimensional skew-symmetric linear differential systems

We prove that the estimates earlier obtained by the authors for the deviations of quasi-integrals of a nonstationary three-dimensional linear differential system with continuously differentiable skew-symmetric coefficient matrix from the corresponding integrals in the stationary case, which are attainable on different nontrivial solutions of the system, are not simultaneously attainable on any single nontrivial solution and at the same time. In the course of proof, we establish new estimates, which are more accurate than those obtained earlier in Mem. Differential Equations Math. Phys., 2007, vol. 41, pp. 157–162. The suggested method for constructing such estimates is also of interest.