Solution of the Helmholtz equation with complex-valued coefficients in open regions

An exact solution is obtained of the boundary-value problems for the Heimholte difference equation in open regions.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 16–21, 1985.     

Inverse spectral problem for a generalized Sturm-Liouville equation with complex-valued coefficients

The present paper is the first to prove that one of the columns of the monodromy matrix and two of the three coefficients (piecewise analytic on the interval [0, 1]) of the equation (f(x)y′)′+(r(x)−λ ² q(x))y = 0 uniquely determine the third coefficient on this interval provided that the values of the functions f(x) and q(x) lie in the lower (or upper) open complex halfplane and on the positive part of the real axis. This unknown coefficient can be reconstructed by finding the unique zero minimum of a specially constructed functional depending on the solutions of the corresponding Cauchy problem and the given elements of the monodromy matrix.     

On the asymptotic behaviour of solution for the generalized double dispersion equation

In this article, we investigate the Cauchy problem for the generalized double dispersion equation in n-dimensional space. We establish the decay estimates of solution to the corresponding linear equation. Under smallness condition on the initial data, we prove the global existence and asymptotic behaviour of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.     

Global solvability tests for a scalar Riccati equation with complex coefficients

We obtain two global solvability tests for a scalar Riccati equation with complex coefficients. One of them is used to prove a test for the existence of a solution of the Redheffer system, which arises when studying a physical model of electromagnetic wave distribution in a transmission line and in a physical model of diffraction of particles along a rod.     

On the asymptotic behaviour of solutions of a certain fourthorder differential equation

In this paper sufficient conditions for the uniform global asymptotic stability of the zero solution of (1.1) are given.     

Some properties of solutions of a scalar Riccati equation with complex coefficients

We give the definition of normal and limit solutions of the Riccati equation with complex coefficients and study some properties of these solutions. We prove the minimality property of a solution of a system of two linear first-order ordinary differential equations.     

Stability of one difference equation with positive coefficients

We consider a linear homogeneous difference equation of ordern with positive coefficients, which is presented in the normal form. We obtain necessary and sufficient conditions for the stability and asymptotic stability. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1276–1280, September, 1990.     

Asymptotic behaviour for wave equation with time-dependent coefficients

Abstract We shall find asymptotic profiles for strictly hyperbolic equations with time-dependent coefficients which are of Lipschitz class and have some stability condition. More precisely, it will be shown that there exists a solution which is not asymptotically free provided that the coefficient tends slowly to some constant. Keywords: Wave equation, Asymptotic profiles, Asymptotic integrations     

On the convergence to zero of solutions of a second-order differential equation with complex-valued potential

We consider the second-order linear differential equation y" + A(t)y = 0 on the semiaxis with complex-valued potential function. Sufficient conditions for the potential function assuring that all solutions of the equation converge to zero at infinity are obtained. It is shown that the conditions imposed on the potential function are close to the necessary ones. One of the results seems to be new even in the case of real-valued function A(·).     

On existence and asymptotic behaviour of solutions of a functional integral equation

The paper contains a result on the existence and asymptotic behaviour of solutions of a functional integral equation. That result is proved under rather general hypotheses. The main tools used in our considerations are the concept of a measure of noncompactness and the classical Schauder fixed point principle. The investigations of the paper are placed in the space of continuous and tempered functions on the real half-line. We prove an existence result which generalizes several ones concerning functional integral equations and obtained earlier by other authors. The applicability of our result is illustrated by some examples.     

On existence and asymptotic behaviour of solutions of a fractional integral equation

We study the solvability of a quadratic integral equation of fractional order with linear modification of the argument. This equation is considered in the Banach space of real functions, defined, bounded and continuous on an unbounded interval. Moreover, we will obtain some asymptotic characterization of solutions.     

On asymptotic properties of solutions defined on a half-axis to one semilinear ordinary differential equation

We consider solutions to the semilinear ordinary differential equation with a nonlinear term of Emden–Fowler type. The results about the asymptotic behavior of the solutions to the Emden–Fowler equation defined in a neighborhood of infinity, presented in the book of R. Bellman, are extended to the case of equation with lower-order derivative.     

On the matrix Riccati equation of transport processes

By keeping count of the number of interactions, we obtain some novel perturbation expansions for the Riccati equation. Similar results were obtained in radiative transfer theory by Mingle.     

On the asymptotic behaviour of Lee-codes

The asymptotic forms of bounds on the information rate of Lee-codes are derived and their relative strength is discussed. Also it is shown that the covering radius of Lee-codes lies asymptotically on the Varshamov-Gilbert bound.     

The asymptotic behaviour of the solution of the filtration equation

It is proved that the self-similar solution of the nonlinear equation of filtration gives the asymptotic representation of the solution of the Cauchy problem for the same equation.     

On the solvability of the Riccati matrix algebraic equation

We obtain conditions of solvability of the Riccati matrix algebraic equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1591–1593, November, 1994.