On the Existence of a Global Solution of a Hyperbolic Problem

Doklady Mathematics - A quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For a simplified...     

关于初值问题解的稳定性

本文利用解对初值和参数的可微性,提出一种不同于Liapunov直接法的方法,用它可以判定初值问题非驻定解的稳定性.而该非驻定解无需事先求出.     

On the Stability of Functional Equations and a Problem of Ulam

In this paper, we study the stability of functional equations that has its origins with S. M. Ulam, who posed the fundamental problem 60 years ago and with D. H. Hyers, who gave the first significant partial solution in 1941. In particular, during the last two decades, the notion of stability of functional equations has evolved into an area of continuing research from both pure and applied viewpoints. Both classical results and current research are presented in a unified and self-contained fashion. In addition, related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.     

On Analyses of the Solution to a Linear Programming Problem

Many complex problem situations in various contexts have been represented in recent years by the linear programming model. The simplex method can then be used to give the optimal values of the variables corresponding to a given set of values of the parameters. However, in many situations it is useful to have the solution to many other related problems which differ from the original problem only in the values of some of the parameters. This paper presents procedures by which the solutions to the changed problems can be derived from the simplex solution tableau corresponding to the original problem. The method will be illustrated by means of an example problem, and it will be shown how quantitative information obtained from such analyses can aid management in decision making.     

A Stability Estimate for a Solution of the Problem of Determining the Dielectric Permittivity

We consider the problem of determining the dielectric permittivity for a nonconducting and nonmagnetic medium. As information we take the traces of the tangential components of the electromagnetic field on the lateral surface of a cylindrical domain. These traces correspond to a solution to some direct problem for the Maxwell system. The impulse source of the current flux lies outside the domain in which the coefficient is sought. The main result of the article is a stability estimate for a solution to the inverse problem in question.     

A Stability Estimate for a Solution to a Two-Dimensional Inverse Problem of Electrodynamics

We consider the problem of finding the three coefficients c(x), (x), and q(x) in a hyperbolic equation. Here c(x) is the coefficient at the Laplace operator, (x) is the coefficient of the first time derivative, and q(x) is the coefficient of the lower-order term. The problem results from the inverse electrodynamic problem of finding the electrodynamic parameters of an isotropic medium under the assumption that the properties of the medium and the exterior current are independent of one coordinate. We suppose that the coefficients c(x)-1, (x), and q(x) are small in some norm and their supports are contained in some disk B. This is equivalent to the assumption that the electrodynamic parameters of the medium are close to constants. We suppose that the source initiating oscillations has the form of the impulse function (t)(x·) localized on the set t=0, x·=0. Here is a unit vector playing the role of a parameter of the problem. The electromagnetic field excited by this source applied outside B is measured at points of the boundary of the domain B on some time interval of a fixed length T counted from the moment of arrival of the signal from the source for three different values of the parameter . It is proven that, for a sufficiently large T, these data determine the sought coefficients uniquely. We obtain a conditional stability estimate for a solution to the problem.     

Global Stability of the Neumann Solution of the Two-phase Stefan Problem

The global stability of the Neumann solution of the two-phaseStefan problem is proved in the framework of the linear theoryof perturbations. As a corollary of this result the spatialstability of the free boundary in a one-dimensional two-phaseCauchy-Stefan problem is proved, assuming that the initial temperaturehas finite limits at infinity, and that its initial perturbationis located in a strip of a finite width.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.

     

关于三阶边值问题解的存在性

利用上下解方法 ,分别讨论了当f∶[0 ,1 ]×R→R在有限区间和无限区间上满足某些增长性条件时 ,三阶微分方程边值问题u (t) +f(t,u) =0 ,u(0 ) =u(1 ) =u″(0 ) =0解与正解的存在性 .最后给出两个例子作为对所获得结果的应用     

On the Approximate Solution of a Diffusion Problem by Galerkin Methods

Discrete-time Galerkin methods are considered for the approximatesolution of a parabolic initial boundary value problem whicharises, for example, in problems involving the diffusion ofa solute into a solid from a stirred solution of fixed volume.Optimal error estimates in the L² and H¹ norms are derived forthe Crank-Nicolson Galerkin method. For the one space variablecase optimal L estimates are also obtained. Results of numericalexperiments are presented and comparisons with finite differenceapproximations are made.     

关于非线性狄立克雷问题的一致有效渐近解

本文研究最高阶导数项带小参数的二阶拟线性椭圆型方程的狄立克雷问题.在退化方程不存在奇点的情形下,当参数ε是充分小时,证明了解的存在性和唯一性,并在整个区域导出解的一致有效渐近近似式.     

关于等周问题级数解法的一些改进

对苏步青教授在文[1]中介绍的改良的Hurwitz方法再作一些改进,先对Wirtinger引理进行推广,再用推广后的Wirtinger不等式很自然而简洁地推出了等周不等式.     

On the Existence of a Generalized Solution of the Saturated-Unsaturated Filtration Problem

We study the existence of a generalized solution of an initial–boundary value problem describing the process of unsteady filtration of a liquid in a bounded region of an n-dimensional space. We consider the case in which the Kirchhoff transformation used to determine the generalized solution takes the real axis into a semiaxis bounded below. An auxiliary problem is constructed. It is proved that any solution of the auxiliary problem is a solution of the problem under study. The solvability of the auxiliary problem is established by using the method of semidiscretization in time and the Galerkin method.     

A Stability Estimate for a Solution to a Three-Dimensional Inverse Problem for the Maxwell Equations

We consider the problem of determining dielectric permittivity and conductivity in the Maxwell equations. As additional information we prescribe the traces of the tangential components of the electromagnetic field on the lateral surface of a cylindric domain. We establish a stability estimate for a solution to the inverse problem and a uniqueness theorem.     

Stability of a Trivial Solution of a Mixed Problem for a Parabolic-Type, Distributed-Parameter System

The Lyapunov stability condition is established for a parabolic-type system with delay. N. N. Krasovskii's idea [1] of using functionals instead of functions for the second method of Lyapunov is borrowed here.     

Estimation of the Solution Stability of the Cauchy Problem with the Data on a Time-Like Plane

Under consideration is the problem of continuation of the wave field from the boundary of a half-plane inside it. We obtain stability estimate for the solution of the corresponding Cauchy problem.     

关于非齐次柯西问题的强解

利用自反Banach空间中弱紧算子的因子分解技巧，对于一类非齐次项具有连续Lipschitz扰动的柯西问题，当其齐次项算子生成强连续算子半群且具有紧豫解式限制时，证明了方程强解的存在性．     

一个离散型淋病数学模型的解的稳定性

本文采用标量李雅普诺夫函数的方法,研究了一个二维离散型淋病数学模型的解的稳定性,同时给出了稳定域的参数估计,并从理论上解释了这个模型的合理性。