A point source of oscillations near a concave mirror. II

The paper is devoted to the construction of a formal shortwave solution of the problem of a point source. The case of a concave reflecting boundary near which the whispering gallery effect occurs is considered. A similar problem was solved earlier for a boundary condition of Dirichlet type. In this work a boundary condition of Neumann type is considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 3–13, 1979.     

Short-wave scattering near the boundry inflection point

We investigate the problem of tangential incidence of short waves onto a surface with an inflection point. Formal solutions of the corresponding equation are constructed near the inflection point in the form of a quasihomogeneous function series. The formal solution is joined with the geometrical optics solution far from the inflection point of the boundary. The problem is restated as a scattering problem for the Schrodinger equation; existence, uniqueness, and smoothness theorems are proved. The formal asymptotic expansions are proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 152–166, 1985.In conclusion, I would like to thank M. M. Popov for suggesting the problem, and also V. M. Babich and M. M. Popov for useful comments.     

Diffraction of short waves on a screen of variable transparency

A model problem of a point source in the field of which there is a screen of variable transparency is considered. The character of the change of transparency of the screen makes it possible to construct an exact solution of the problem by the method of separation of variables. A shortwave asymptotic expression for the wave field is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 24–32, 1981.In conclusion, the author thanks V. A. Borovikov for posing the problem and V. M. Babich for assistance with the work.     

Short-wavelength asymptotic solution of the problem of a moving point source in a time-dependent inhomogeneous medium

A uniform short-wavelength asymptotic solution is obtained for the problem of the field of a point source moving with subsonic velocity in a time-dependent inhomogeneous medium; the solution is valid both in a small neighborhood of the source and at a large distance from it.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 42–51, 1987.     

Asymptotic Solution of the Autoresonance Problem

We study the problem of forced oscillations near a stable equilibrium of a two-dimensional nonlinear Hamiltonian system of equations. A given exciting force is represented as rapid oscillations with a small amplitude and a slowly varying frequency. We study the conditions under which such a perturbation makes the phase trajectory of the system recede from the original equilibrium point to a distance of the order of unity. To study the problem, we construct asymptotic solutions using a small amplitude parameter. We present the solution for not-too-small values of time outside the original boundary layer.     

Solution of the skin effect problem with an arbitrary coefficient of specular reflection

An analytical solution of the skin effect problem in a metal with specular-diffuse boundary conditions is obtained. A new analytical method is developed that makes it possible to obtain a solution up to an arbitrary degree of accuracy. The method is based on the idea of representing not only the boundary condition on the field in the form of a source (which is conventional) but also the boundary condition on the distribution function. The solution is obtained in the form of a von Neumann series.     

On singular solutions of the initial boundary value problem for the Stokes system in a power cusp domain

The initial boundary value problem for the non-steady Stokes system is considered in bounded domains with the boundary having a peak-type singularity (power cusp singularity). The case of the boundary value with a nonzero time-dependent flow rate is studied. The formal asymptotic expansion of the solution near the singular point is constructed. This expansion contains both the outer asymptotic expansion and the boundary-layer-in-time corrector with the ‘fast time’ variable depending on the distance to the cusp point. The solution of the problem is constructed as the sum of the asymptotic expansion and the term with finite energy.     

On a point source of harmonic oscillations in a homogeneous anisotropic elastic medium

A point source of harmonic oscillations in an infinite homogeneous anisotropic elastic space is considered. It is shown that everywhere, except for certain directions, the phase function of the problem solution can be determined by applying the Legendre transform to the characteristic function of the equations. The group velocity of the solution is directed from the point source precisely along the radius. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 14–20. Translated by N. S. Zabavnikova.     

On numerical study of periodic solutions of a delay equation in biological models

Some results are presented of the numerical study of periodic solutions of a nonlinear equation with a delayed argument in connection with themathematical models having real biological prototypes. The problem is formulated as a boundary value problem for a delay equation with the conditions of periodicity and transversality. A spline-collocation finite-difference scheme of the boundary value problem using a Hermitian interpolation cubic spline of the class C ¹ with fourth order error is proposed. For the numerical study of the system of nonlinear equations of the finitedifference scheme, the parameter continuation method is used, which allows us to identify possible nonuniqueness of the solution of the boundary value problem and, hence, the nonuniqueness of periodic solutions regardless of their stability. By examples it is shown that the periodic oscillations occur for the parameter values specific to the real molecular-genetic systems of higher species, for which the principle of delay is quite easy to implement.     

New closed-form Green function and integral formula for a thermoelastic quadrant

In this study a new Green’s function and a new Green-type integral formula for a boundary value problem (BVP) in thermoelastostatics for a quadrant are derived in closed form. On the boundary semi-straight-lines twice mixed homogeneous mechanical boundary conditions (one boundary semi-straight-line is free of loadings and normal displacements and tangential stresses are prescribed on the other one) are prescribed. The thermoelastic displacements are subject by a heat source applied in the inner points of the quadrant and by mixed non-homogeneous boundary heat conditions (on one boundary semi-straight-line the temperature is prescribed and the heat flux is given on the other one). When thermoelastic Green’s function is derived the thermoelastic displacements are generated by an inner unit point heat source, described by δ Dirac’s function. All results are obtained in elementary functions that are formulated in a special theorem. A closed-form solution for a particular BVP of thermoelastostatics for a quadrant also is included. Using the proposed approach it is possible to extend the obtained for quadrant results to any other canonical Cartesian domain.     

An inequality for the reduced wave operator and the justification of geometrical optics

We present an inequality for the reduced wave operator in the exterior of a star-shaped surface in n-space, with a Dirichlet boundary condition on the surface and a radiation condition at infinity. This inequality is used to demonstrate the continuous dependence (in a suitable norm) of the solution of a scattering problem upon the boundary data and inhomogeneous term in the differential equation. This basic result is then used together with the results of D. Ludwig [7] to prove that the formal solution of the scattering problem for a convex body, which is given by geometrical optics, is asymptotic to the exact solution. Similar results have been given in two dimensions by V. S. Buslaev [1] and R. Grimshaw [2], using different methods, who also consider the Neumann problem. Unfortunately the methods used here are inapplicable in that case.     

A rigorous proof of the asymptotic expansion of Green's function in the shadow zone for diffraction on a convex body

We prove the asymptotic expansion (for is the wavenumber) of Green's function for diffraction on a convex smooth body, with both the source and the observation point at an arbitrary distance from the boundary. The analysis is conducted for a two-dimensional Dirichlet problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 79–88, 1985.I acknowledge the useful comments of V. B. Filippov.     

A point source of oscillations on the boundary of a region

The problem of a point source of oscillations on a curveS with curvature which is nowhere zero is considered: The case where the whispering gallery effect arises is investigated: rays issuing from a source and reflected many times fromS create this effect. A function containing all singularities ofw is constructed explicitly. The theorem that the set of singularities of the functionw coincides with the wave fronts of geometrical optics is a consequence of these considerations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 3–20, 1976.     

Features of the dynamic stresses inthe neighbourhood of the joint of three elastic media

The solution of the problem of the harmonic oscillations of a piecewise-inhomogeneous domain, consisting of three joinedrectangles with different elastic properties, is constructed within the framework of a modified superposition method. The discontinuities in the wave field are investigated in the neighbourhood of singular point of the boundary at the joint of the rectangles.     

Classification of Harmonic Functions in the Exterior of the Unit Ball

We solve the Laplace equation in an exterior infinite spherical domain with nonlinear (quadratic) boundary conditions on the spherical boundary. We linearize the problem and, under the additional assumption that the distinguishing function is spherically symmetric, write the solution by using the formal power series method with recursion of the series coefficients. Applying the Poincaré--Perron theorem, we describe the space of convergent formal power series and calculate its dimension. Estimating the roots of the fourth-degree characteristic polynomial corresponding to the given problem, we also calculate the dimension of the space of functions whose gradient at each point of the sphere is orthogonal to the linear combination of an axially symmetric dipole and a quadrupole. In conclusion, we state several unsolved problems arising in geophysical applications.     

A uniqueness theorem in the inverse problem for the integrodifferential electrodynamics equations

Under study is the problem of finding the kernel and the index of dielectric permeability for the system of integrodifferential electrodynamics equations with wave dispersion. We consider a direct problem in which the external pulse current is a dipole located at a point y on the boundary ?B of the unit ball B. The point y runs over the whole boundary and is a parameter of the problem. The information available about the solution to the direct problem is the trace on ?B of the solution to the Cauchy problem given for the times close to the time when a wave from the dipole source arrives at a point x. The main result of the article consists in obtaining some theorems related to the uniqueness problems for a solution to the inverse problem.     

Fourier method in a mixed problem for the wave equation on a graph

A mixed problem for the wave equation on the simplest geometric graph consisting of two ring edges that touch at a point is considered. The approach used is based on the contour integration of the operator’s resolvent. With the help of a special transformation of a formal series, a classical solution of the problem is obtained under minimum conditions imposed on the initial data. This approach makes it possible to do without an expensive analysis of improved asymptotics for the eigenvalues and eigenfunctions of the operator and to avoid the difficulties associated with the possible multiplicity of the operator’s spectrum.     

Behavior of the generalized solutions of the Dirichlet problem for higher-order elliptic equations in the neighborhood of the boundary

Making use of the method of weight functions and of energy inequalities, similar to the Saint-Venant principle, the authors obtain estimates which characterize the behavior of the generalized solutions of the Dirichlet problem for the general higher-order elliptic equation in the neighborhood of a boundary point. In the case of two independent variables one has obtained an estimate of the maximum of the modulus of the solution in the neighborhood of a boundary point.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 114–125, 1982.     

Asymptotic Integration of Parabolic Problems with Large High-Frequency Summands

We develop the averaging method theory for parabolic problems with rapidly oscillating summands some of which are large, i.e., proportional to the square root of the frequency of oscillations. In this case the corresponding averaged problems do not coincide in general with those obtained by the traditional averaging, i.e., by formally averaging the summands of the initial problem (since the principal term of the asymptotic expansion of a solution to the latter problem is not in general a solution to the so-obtained problem). In this article we consider the question of time periodic solutions to the first boundary value problem for a semilinear parabolic equation of an arbitrary order 2k whose nonlinear terms, including the large, depend on the derivatives of the unknown up to the order k-1. We construct the averaged problem and the formal asymptotic expansion of a solution. When the large summands depend on the unknown rather than its derivatives we justify the averaging method and the complete asymptotic expansion of a solution.Original Russian Text Copyright © 2005 Levenshtam V. B.The author was supported by the Russian Foundation for Basic Research (Grant 01-01-00678) and the Program “ Universities of Russia” (UR.04.01.029).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 805–821, July–August, 2005.     

Convergence with orderh 2p-1 of a 2p+1-point scheme of the method of lines for one boundary-value problem

A multipoint scheme of the method of lines is applied to the boundary-value problem The second derivative replaced by a-point ( is any positive integer) central-difference approximation with an error of order where is the step of the net of lines. An approximating system of ordinary differential equations, associated with problem (1), (2), is transformed into a reducing one. The uniform convergence of the approximate solution of the method of lines to the solution of the original boundary-value problem with order is established. For this the solution of the reducing system with zero boundary conditions is examined for the difference between the exact solution of problem (1), (2) and the approximate solution obtained by the method of lines. The behavior of this solution as is studied at point and, next, at any point by transforming the independent variable that transfers point z to the origin.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 39–45, 1979.