Boundary Integral Equation Method in the Steady State Oscillation Problems for Anisotropic Bodies

The three-dimensional steady state oscillation problems of the elasticity theory for homogeneous anisotropic bodies are studied. By means of the limiting absortion principle the fundamental matrices maximally decaying at infinity are constructed and the generalized Sommerfeld–Kupradze type radiation conditions are formulated. Special functional spaces are introduced in which the basic and mixed exterior boundary value problems of the steady state oscillation theory have unique solutions for arbitrary values of the oscillation parameter. Existence theorems are proved by reduction of the original boundary value problems to equivalent boundary integral (pseudodifferential) equations. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

一类边值问题的振动准则

本文考虑一类中立型泛函偏微分方程的边值问题。我们的主要工具是平均技巧，利用有关的泛函微分不等式来建立这类边值问题解的振动准则。     

Eigenvalues of second-order linear equations with coupled boundary conditions on time scales

In this paper we consider coupled boundary value problems for second-order linear equations on time scales. By properties of eigenvalues of the Dirichlet boundary value problem and some oscillation results, existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only extend the existing ones of coupled boundary value problems for second-order difference equations but also contain more complicated time scales.     

Boundary value problems in the theory of binary mixtures

In this paper, the boundary value problems of steady oscillation (vibration) of the linear theory of thermoelasticity for binary mixtures are investigated by means of the boundary integral equation method (potential method). The uniqueness and existence theorems of solutions of the exterior boundary value problems by means potential method and multidimensional singular integral equations are proved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic distribution of the eigenvalues and eigenfunctions in basic boundary value oscillation problems in hemitropic elasticity

The basic boundary value oscillation problems for a three-dimensional elastic medium bounded by a closed surface are considered. Asymptotic formulas are derived for the eigenvalue and eigenfunction distributions in the problems.     

具有连续时滞的中立型双曲偏微分方程系统解的振动性

研究了一类具有连续时滞变量的非线性中立型双曲型偏微分方程系统,解的振动性.获得了该方程组在Robin边值条件和Dirichlet边值条件下解振动的充分条件.     

Applications of maximum principles to dynamic equations on time scales

We apply the strong maximum principle to obtain a priori bounds and uniqueness of solutions for some initial value and boundary value problems as well as to establish oscillation results for second-order dynamic equations on time scales. Our comparison, uniqueness, and oscillation results are new and are extensions of results for ordinary differential equations to the times scale setting.     

一类具有连续分布的偏差变元的双曲型方程的振动准则

本文研究了连续分布偏差变元的双曲型方程，利用化多维振动问题为一维问题的方法，得到了某些边值问题的每一个解均为振动的充分条件。所得条件推广和改进了文献中的结果。     

Oscillation criteria for boundary value problems of high-order partial functional differential equations

In this paper, a class of boundary value problems associated with high-order partial functional differential equations with distributed deviating arguments is investigated. Some oscillation criteria of solutions to the problem are developed. Our approach is to reduce the multi-dimensional oscillation problem to a one-dimensional oscillation one by employing some integral means of solutions and introducing some parameter functions. One illustrative example is considered.     

Oscillation of even order partial differential equations with distributed deviating arguments

Some Kamenev-type oscillation criteria are established for a class of boundary value problems associated with even-order partial differential equations with distributed deviating arguments. Our approach is to reduce the high-dimensional oscillation problem to a one-dimensional oscillation one, and the general means developed by Philos and Wong is used as the main tool. The results obtained here extend and improve some known results in the literature.     

Smoothness properties of Newtonian potentials in the study of elastic plates

The Hölder continuity and differentiability are investigated of Newtonian potentials arising in the theory of bending of elastic plates with transverse shear deformation. These properties play an essential role in the study by means of boundary integral equation techniques of boundary value problems for the equilibrium and harmonic oscillation states, and in the construction of associated boundary element methods.     

一类非线性偏泛函微分方程的强迫振动性

讨论了一类时滞双曲方程边值问题,给出了该类方程在三类边界条件下解的振动条件.     

Oscillations of certain nonlinear delay parabolic boundary value problems

In this paper we consider some nonlinear parabolic partial differential equations with distributed deviating arguments and establish sufficient conditions for the oscillation of some boundary value problems.     

Asymptotic behavior of eigenfunctions and eigenfrequencies of oscillation boundary value problems of the linear theory of elastic mixtures

The asymptotic behavior of eigenoscillation and eigen-vector-function is studied for the internal boundary value problems of oscillation of the linear theory of a mixture of two isotropic elastic media.     

Interaction between thermoelastic and scalar oscillation fields

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields in a general physically anisotropic case are studied by the boundary integral equation methods. Uniqueness and existence theorems are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations. In the non-resonance case the invertibility of the corresponding matrix pseudodifferential operators in appropriate functional spaces is shown on the basis of the generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case the co-kernels of the pseudodifferential operators are analysed and the efficient conditions of solvability of the original interface problems are established.     

Oscillation of solutions of parabolic differential equations of neutral type

In this paper, we investigate a class of parabolic differential equations of neutral type, and obtain some sufficient conditions of the oscillation for such equations satisfying two kinds of boundary value conditions.     

On the Kolodner-Coffman method for the uniqueness problem of Emden-Fowler BVP

This paper surveys the Kolodner-Coffman method which has been applied very successfully by many authors to obtain uniqueness theorems for boundary value problems for differential equations of Emden-Fowler type. We recast some of the arguments in the language of Sturm's oscillation theory for linear second-order differential equations. The clarification provided by the new perspective enables us to improve several known results.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38.     

非线性二阶时滞微分不等式的性质及其应用

本文研究一类非线性二阶时滞微分不等式解的性质。应用这些性质,建立了一类含时滞的双曲偏微分方程边值问题解的若干新的振动准则。     

Sturm theory for the equation of vibrating beam

In this paper we develop an extension of the classical Sturm theory [C. Sturm, Sur une classe d'equations à derivée partielle, J. Math. Pures Appl. 1 (1836) 373-444], to study the oscillation properties for the eigenfunctions of some fourth-order two point boundary value problems on the interval [0,1]. We are mainly interested in the case when these problems have negative eigenvalues induced by the sign of the parameters in the boundary conditions. In particular, we give an asymptotic estimate of the number of zeros in (0,1) of the first eigenfunction in terms of the variation of parameters in the boundary conditions.