Application of nonclassical separation of variables to a nonlocal heat problem

We consider an initial-boundary value problem for the heat equation with a nonlocal two-point boundary condition containing a parameter. By separating the variables in an auxiliary function system, we construct a regular solution. We obtain sufficient conditions for the absolute and uniform convergence of the series in the auxiliary system. We prove conditions close to necessary ones for the existence of a regular solution of the initial-boundary value problem.     

The dressing method and separation of variables: The two-dimensional case

Separation of variables is very convenient for obtaining solutions of linear differential equations in explicit form. We use the dressing method to widen the class of such equations. As an example, we dress a two-dimensional linear differential operator, including an operator with constant coefficients.     

Axially symmetric problem of local separation of an elastic half-space from a rigid base due to a point source of cooling

An axially symmetric problem on the unilateral frictionless contact of an elastic half-space and a rigid thermally insulated base under the action of a subsurface source of cooling, which leads to local loss of contact over a circular region, is considered. The intensity of the source at which the local loss of contacts begins is determined. The dependence of the geometrical parameters of the gap between the surfaces of the bodies on the intensity of the source of cooling and its location relative to the surface is analyzed in detail.     

On the solution of a nonlinear stationary inhomogeneous filtration problem with a point source

We prove the convergence of the simple iteration method at the rate of a geometric progression in the norm of Sobolev and Hölder spaces to the solution of a stationary nonlinear inhomogeneous filtration problem with a point source.     

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.     

Short-wave asymptotics of the solution of the problem of a point source in an inhomogeneous moving medium

The leading term is obtained for the short-wave asymptotics of the field of a point source in a small neighborhood of it and also an expression for the excitation coefficient of the wave in the region where the ray representation is valid for an infinite, inhomogeneous, vortex-free, moving medium with constant entropy and for a weakly vortical medium for which the speed of motion is small.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 41–48, 1984.The author is grateful to V. S. Buldyrev for useful discussion of the papers.     

New variables of separation for particular case of the Kowalevski top

We discuss the polynomial bi-Hamiltonian structures for the Kowalevski top in special case of zero square integral. An explicit procedure to find variables of separation and separation relations is considered in detail.     

The streamline–diffusion method for a convection–diffusion problem with a point source

A linear singularly perturbed convection–diffusion problem with a point source is considered. The problem is solved using the streamline–diffusion finite element method on a class of Shishkin–type meshes. We prove that the method is almost optimal with uniform second order of convergence in the maximum norm. We also prove the existence of superconvergent points for the first derivative. Numerical experiments support these theoretical results.     

The point source field near a concave boundary

A new representation for the field emitted by a short-wave source placed near a concave boundary is obtained by the hybrid method. A Dirichlet problem is considered. The representation obtained provides an efficient algorithm for numerical experiments. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 212–219.     

Diffraction of the field of a point source by a circular cone (nonaxisymmetric case)

The problem of the diffraction of the wave field of a point source by an ideally reflecting circular cone is considered; the source is located off the axis of the cone but at a distance from it such that the space outside the cone is completely illuminated by waves from the source. Equations are obtained describing the behavior of the diffraction field in various regions of variation of the angular coordinates and which are convenient for numerical computations; integrals corresponding to the field of the incident and reflected waves are aso distinguished.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 212–227, 1974.     

From Jacobi problem of separation of variables to theory of quasipotential Newton equations

Our solution to the Jacobi problem of finding separation variables for natural Hamiltonian systems H = ½p ² + V(q) is explained in the first part of this review. It has a form of an effective criterion that for any given potential V(q) tells whether there exist suitable separation coordinates x(q) and how to find these coordinates, so that the Hamilton-Jacobi equation of the transformed Hamiltonian is separable. The main reason for existence of such criterion is the fact that for separable potentials V(q) all integrals of motion depend quadratically on momenta and that all orthogonal separation coordinates stem from the generalized elliptic coordinates. This criterion is directly applicable to the problem of separating multidimensional stationary Schrödinger equation of quantum mechanics. Second part of this work provides a summary of theory of quasipotential, cofactor pair Newton equations $ \ddot q $ = M(q) admitting n quadratic integrals of motion. This theory is a natural generalization of theory of separable potential systems $ \ddot q $ = ??(q). The cofactor pair Newton equations admit a Hamilton-Poisson structure in an extended 2n + 1 dimensional phase space and are integrable by embedding into a Liouville integrable system. Two characterizations of these systems are given: one through a Poisson pencil and another one through a set of Fundamental Equations. For a generic cofactor pair system separation variables have been found and such system have been shown to be equivalent to a Stäckel separable Hamiltonian system. The theory is illustrated by examples of driven and triangular Newton equations.     

Behavior of wave fields in the problem of diffraction of waves from a point source by an ideally reflecting,circular cone (axisymmetric case)

Wave fields represnting the solution of the problem of stationary diffraction of monochromatic spherical waves by a conical region with zero boundary conditions are studied. Approximate forms of solutions convenient for investigation in various regions of variation of the independent variables are presented in the hf approximation. Graphs are constructed on the basis of numerical computations, and the ensuing consequences are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 192–196, 1976.     

On one boundary value problem for a nonlinear heat equation in the case of two space variables

The article addresses a boundary value problem with degeneration for a nonlinear heat equation in the case of two space variables. Solving this problem makes it possible to study heat conduction in a neighborhood of a closed cylindrical surface. The theorem of the existence and uniqueness of an analytic solution to the problem is proved.     

Justification of a construction of the solution of the problem on a point source ofSH waves in a layered medium with vertical boundary

The solution is determined by the method of separation of variables. The method is justified by using a suitable space of generalized functions. Boundary conditions “are hidden” in this space. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 56–66. Original article This work was supported by the Russian Foundation of Fundamental Research (Grant No. 93-011-16148). Translated by T. N. Surkova.     

A two dimensional thermoviscoelastic problem due to instantaneous point heat source

A dynamic problem of thermoviscoelasticity in the presence of heat source is studied here. The solution in Laplace–Fourier transformed domain is obtained using eigen value approach. Stress distributions are computed numerically.     

The transfer point location problem

In this paper, we introduce the transfer point location problem. Demand for emergency service is generated at a set of demand points who need the services of a central facility (such as a hospital). Patients are transferred to a helicopter pad (transfer point) at normal speed, and from there they are transferred to the facility at increased speed. The general model involves the location of p helicopter pads and one facility. In this paper, we solve the special case where the location of the facility is known and the best location of one transfer point that serves a set of demand points is sought. Both minisum and minimax versions of the models are investigated. In follow up papers we investigate the general model using the results obtained in this paper.     

The affine separation problem revisited

The aim of this note is to characterize in terms of inequalities those pairs of real functions (acting on a convex subset of a vector space) that possess an affine separator. The main result is originally due to Behrends and Nikodem. Their method is based on the Hahn–Banach Theorem and a variant of the Helly Theorem. In our approach, a direct and independent proof is presented via the Radon and the Helly Theorems.