The solvability of integral lightguide equations

Integral equations of a three-dimensional periodic lightguide are studied for arbitrary values of the frequency parameter. The solution to the homogeneous integral equation vanishes for those values of the spectral parameter that are different from some isolated points. Bibliography: 5 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 89–100.     

Integral equations as evolution equations

We study a class of time-dependent linear integrodifferential equations (VE) with the evolution equation approach. We determine the generators of a time-dependent evolution equation (DE) which is equivalent to the given integrodifferential equation. Under very general assumptions we prove the well-posedness and continuity of (VE) from the stability of (DE). The related question of convergence of a family of approximate solutions is examined. As an application, we include an example of hyperbolic integro-partial-differential equation to illustrate the theory.     

Integral equations requiring small numbers of Krylov-subspace iterations for two-dimensional smooth penetrable scattering problems

This paper presents a class of boundary integral equations for the solution of problems of electromagnetic and acoustic scattering by two-dimensional homogeneous penetrable scatterers with smooth boundaries. The new integral equations, which, as is established in this paper, are uniquely solvable Fredholm equations of the second kind, result from representations of fields as combinations of single and double layer potentials acting on appropriately chosen regularizing operators. As demonstrated in this text by means of a variety of numerical examples (that resulted from a high-order Nyström computational implementation of the new equations), these “regularized combined equations” can give rise to important reductions in computational costs, for a given accuracy, over those resulting from previous iterative boundary integral equation solvers for transmission problems.     

Integral equations of Volterra type

The behavior of exact solutions to Volterra linear and non-linear integral equations with negative or positive, monotone kernels is studied. It includes properties such as the number of zeroes, boundedness and monotonicity of the solutions on the infinite interval.     

Invariant subspaces of a lightguide with periodic boundary

A waveguide operator is defined. It is proved that its spectrum coincides with the spectrum of a lightguide. The classification of singular points of the continuous spectrum is given. Invariant subspaces of the waveguide operator are distinguished that are related to an interval of the continuous spectrum without singular points. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 51–62.     

Integral equations and sound propagation in a shallow sea

The boundary value problem for the Helmholtz equation for the acoustic pressure in a shallow sea can be reduced to a system of integral equations (some of which can be hypersingular). We suggest a numerical method for solving this system. In this approach to the numerical solution of the sound propagation problem in a shallow sea, the surfaces of the sea, the seabed, and the layers can have an arbitrary geometric structure.Translated from Differentsialnye Uravneniya, Vol. 40, No. 9, 2004, pp. 1256–1270.Original Russian Text Copyright © 2004 by Lifanov, Stavtsev.     

Integral sets of a class of differential equations with impulses

The problem of the existence and stability of an invariant set of a class of systems of differential equations with impulses is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 994–997, July, 1990.     

Integral invariants of the Hamilton equations

Conditions are found for the existence of integral invariants of Hamiltonian systems. For two-degrees-of-freedom systems these conditions are intimately related to the existence of nontrivial symmetry fields and multivalued integrals. Any integral invariant of a geodesic flow on an analytic surface of genus greater than 1 is shown to be a constant multiple of the Poincaré-Cartan invariant. Poincaré's conjecture that there are no additional integral invariants in the restricted three-body problem is proved. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 379–393, September, 1995. The work was financially supported by the Russian Foundation for Basic Research (grant No. 242 93-013-16244), International Science Foundation (grant No. MCY 000), and INTAS (grant No. 93-339).     

Integral sets of systems of difference equations

The problem of existence of integral sets of systems of difference equations is studied. We establish sufficient conditions for the existence of these sets and their stability. For the system under consideration, the behavior of trajectories that originate in a sufficiently small neighborhood of integral sets is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1613–1621, December, 1993.     

Integral equations with non integrable kernels

We study here some integral equations linked to the Laplace or the Helmholtz equation, or to the system of elasticity equations. These equations lead to non integrable kernels only defined as finite parts, so that they are quite difficult to approximate. In each case, we introduce a variational formulation which avoids this difficulty and allow us to use stable finite element approximations for these problems     

Reduction of a multidimensional poincare-invariant nonlinear equation to two-dimensional equations

We study the structure of the invariants of the extended isochronous Galilean algebra which is a subalgebra of the Poincaré algebra AP(1, n). Using these results we classify maximal subalgebras of rank n–2 and n–1 of AP(1, n). With respect to subalgebras of rank n–1 of AP(1, n) ansatzes are constructed reducing the equation (u, (u)², u)=0 to differential equations in two invariant variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1311–1323, October, 1991.