Nonhomogeneous String Research Articles

Uniformly exponentially stable approximations for a class of second order evolution equations

We consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one dimensional vibrating systems or of square plates. It is by now well known that the approximating systems obtained by usual finite element or finite difference are not, in general, uniformly stable with respect to the discretization parameter. Our main result shows that, by adding a suitable numerical viscosity term in the numerical scheme, our approximations are uniformly exponentially stable. This result is then applied to obtain strongly convergent approximations of the solutions of the algebraic Riccati equations associated to an LQR optimal control problem. We next give an application to a non-homogeneous string equation. Finally we apply similar techniques for approximating the equations of a damped square plate.

Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions

This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with x-dependent coefficients u(x)ytt – (u(x)yx)x + g(x,t,y) = f(x,t) on (0,π) × ℝ under the periodic boundary conditions y(0,t) = y(π,t), yx(0,t) = yx(π,t) or anti-periodic boundary conditions y(0, t) = –y(π,t), yx[0,t) = – yx(π,t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the ‘weak solution’. For T, the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

Periodic solutions to one-dimensional wave equation with x-dependent coefficients

In this paper, we study the nonlinear one-dimensional periodic wave equation with x-dependent coefficients u ( x ) y t t − ( u ( x ) y x ) x + g ( x , t , y ) = f ( x , t ) on ( 0 , π ) × R under the boundary conditions a 1 y ( 0 , t ) + b 1 y x ( 0 , t ) = 0 , a 2 y ( π , t ) + b 2 y x ( π , t ) = 0 ( a i 2 + b i 2 ≠ 0 for i = 1 , 2 ) and the periodic conditions y ( x , t + T ) = y ( x , t ) , y t ( x , t + T ) = y t ( x , t ) . Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. A main concept is the notion “weak solution” to be given in Section 2. For T is the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

The conditions for application of Fourie method to the solution of nonhomogeneous string oscillation equation with <I>ϕ</I>-subgaussian right side

The conditions for application of Fourie method to the solution of nonhomogeneous string oscillation equation with <I>ϕ</I>-subgaussian right side

The conditions for application of Fourie method to the solution of nonhomogeneous string oscillation equation with φ-subgaussian right side

A nonhomogeneous string oscillation equation with trivial initial and boundary conditions and φ-subgaussian right side is considered. The conditions are found, under which the Fourie method can be applied to this problem, that is the solution of this problem can be represented in the form of uniformly convergent in probability series.

Geometric properties of root vectors for equation of nonhomogeneous damped string: transformation operators method

The spectral decomposition theorem for a class of nonselfadjoint operators in a Hilbert space is obtained in the paper. These operators are the dynamics generators for the systems governed by 1–dim hyperbolic equations with spatially nonhomogeneous coefficients containing first order damping terms and subject to linear nonselfadjoint boundary conditions. These equations and boundary conditions describe, in particular, a spatially nonhomogeneous string subject to a distributed viscous damping and also damped at the boundary points. The main result leading to the spectral decomposition is the fact that the generalized eigenvectors (root vectors) of the above operators form Riesz bases in the corresponding energy spaces. The proofs are based on the transformation operators method. The classical concept of transformation operators is extended to the equation of damped string. Originally, this concept was developed by I. M. Gelfand, B. M. Levitan and V. A. Marchenko for 1–dim Schrödinger equation in connection with the inverse scattering problem. In the classical case, the transformation operator maps the exponential function (stationary wave function of the free particle) into the Jost solution of the perturbed Schrödinger equation. For the equation of a nonhomogeneous damped string, it is natural to introduce two transformation operators (outgoing and incoming transformation operators). The terminology is motivated by an analog with the Lax—Phillips scattering theory. The transformation operators method is used to reduce the Riesz bases property problem for the generalized eigenvectors to the similar problem for a system of nonharmonic exponentials whose complex frequencies are precisely the eigenvalues of our operators. The latter problem is solved based on the spectral asymptotics and known facts about exponential families. The main result presented in the paper means that the generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of non—selfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems.

A classical approach to eigenvalue problems associated with apair of mixed regular Sturm‐Liouville equations II

In the studies of acoustic waveguides in ocean, buckling of columns with variable cross sections in applied elasticity, transverse vibrations in nonhomogeneous strings, etc., we encounter a new class of problems of the type defined on an interval [d1, d2] and defined on the interval [d2, d3] satisfying certain matching conditions at the interface point x = d2.Earlier, in Part I, we constructed a fundamental system for (L1, L2) and derived certain estimates for the same.Here, in Part II, we consider four types of boundary value problems associated with (L1, L2) and study the corresponding spectra.

On Local Solutions of a Mildly Degenerate Hyperbolic Equation

In this work we prove the existence of a unique nonlocal solution for the vibrations of a nonhomogeneous stretched string in Sobolev spaces and without any smallness conditions on the size of the initial data.

An interpolation problem in the class of stieltjes functions and its connection with other problems

This paper contains the proofs and some development of results that were published without proofs in [KN2]. It is completed with comments added by the second author explaining how these results became the basis of statements and the solutions of problems in the theory of entire functions, in the moment problem, in direct and inverse problems of the spectral theory of nonhomogeneous strings and in other problems.

Spectral operators generated by damped hyperbolic equations

We announce a series of results on the spectral analysis for a class of nonselfadjoint opeators, which are the dynamics generators for the systems governed by hyperbolic equations containing dissipative terms. Two such equations are considered: the equation of nonhomogeneous damped string and the 3-dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. Nonselfadjoint boundary conditions are imposed at the ends of a finite interval or on a sphere centered at the origin respectively. Our main result is the fact the aforementioned operators are spectral in the sense of N. Dunford. The result follows from the fact that the systems of root vectors of the above operators form Riesz bases in the corresponding energy spaces. We also give asymptotics of the spectra and state the Riesz basis property results for the nonselfadjoint operator pencils associated with these operators.

Exact Controllability of the Damped Wave Equation

We study the controllability problem for a distributed parameter system governed by the damped wave equation $$ u_{tt}-\frac{1}{\rho(x)}\frac{d}{dx}\left(p(x)\frac{du}{dx}\right)+ 2d(x)u_t+q(x)u=g(x)f(t), $$ where $x\in (0,a)$, with the boundary conditions $$ u(0)=0,\;\; (u_x+hu_t)(a)=0,\;\; h\in{\Bbb C}\cup\{\infty\}. $$ This equation describes the forced motion of a nonhomogeneous string subject to a viscous damping with the damping coefficient $d(x)$ and with damping (if Re $h>0$) or energy production (if Re $h<0$) at one end. (All results extend to the case when a similar condition is imposed at the other end as well.) The function $f(t)$ is considered as a control. Generalizing well-known results by D. Russell concerning the string with $d(x)=0$, we give necessary and sufficient conditions for exact unique controllability and approximate controllability of the system. Our proofs are based on recent results by M. Shubov concerning the spectral analysis of a class of nonselfadjoint operators and operator pencils generated by the above equation.

Computation of Green′s matrices for boundary value problems associated with a pair of mixed linear regular ordinary differential operators

An algorithm for the computation of Green′s matrices for boundary value problems associated with a pair of mixed linear regular ordinary differential operators is presented and two examples from the studies of acoustic waveguides in ocean and transverse vibrations in nonhomogeneous strings are discussed.

Fundamental systems and solutions of nonhomogeneous equations for a pair of mixed linear ordinary differential equations

AbstractTwo different ordinary differential operators L1 and L2 (not of the same order) defined on two adjacent intervals I1 and I2, respectively, with certain mixed conditions at the interface are considered. These problems are encountered in the study of ‘acoustic waveguides in ocean’, ‘transverse vibrations in nonhomogeneous strings’, etc. A complete set of physical conditions on the system give rise to three types of (selfadjoint) boundary value problems associated with the pair (L1, L2). In a series of papers, a systematic study of these new classes of problems is being developed. In the present paper, we construct the fundamental systems and exhibit the forms of solutions of nonhomogeneous problems associated with the pair (L1, L2).

On languages with non-homogeneous strings of quantifiers

It is shown that every linear string of quantifiers can be replaced by a well-ordered sequence of quantifiers.

Some results on the frequencies of nonhomogeneous rods

In this paper we consider the transverse vibrations of a thin cylindrical rod of variable density and constant f lexural r igidity which is clamped at both ends. In Section I we prove two results on the principal frequency of these rods. These theorems are analogous to theorems on the principal frequency of nonhomogeneous strings. Theorem 1 corresponds to a result of Beesack [1] and Theorem 2 to a result of Krein [2]. In Section I I we consider nonhomogeneous clamped rods with equimeasurable density and we prove an inequality for the min imum of the nth frequency (n =2~). This Theorem 3 should be compared with a much stronger statement on the extrema of the frequencies of equimeasurable strings. We rely heavily upon results of Beesack [1, 3]. Notation and methods are similar to those used in our paper on nonhomogeneous strings [4].

On the least positive eigenvalue of integral equations with equimeasurable kernels

Rearrangements of sets of numbers and rearrangements of functions were defined and investigated in detail in the book of Hardy, Littlewood and Polya [4, Chapter X]. Using this notion, classes of nonhomogeneous strings, membranes, rods and plates with equimeasurable density were considered by P. R. Beesack and the author and the extrema of the principal frequency were found for these classes [1; 2; 7]. Here we deal with an analogous question for integral equations. For any equimeasurable class of non-negative and symmetric L2 kernels we find the maximum of the first eigenvalue. The proof is elementary and uses only the maximum property of the reciprocal of the first eigenvalue [3; 8; 9] and a simple geometric idea. After the proof of this theorem we add some remarks on the corresponding minimum and on the same problem for matrices.

Sur la fréquence fondamentale d'une membrane vibrante: évaluations par défaut et principe de maximum

The starting point of the present paper is a simple method ofPayne-Weinberger to obtain lower bounds for the first eigenvalue of a membrane through comparison with the Rayleigh principle for homogeneous strings. A generalization of this idea to auxiliary non-homogeneous strings leads to a maximum principle (7) (of which an inequality ofBarta-Polya is a special case) or (7′); in its form (8), it was essentially known (although with some unnecessary restrictive hypotheses) toM. H. Protter in 1958. — This principle is closely related to theThomson principle of boundary value problems; in particular, the allowed discontinuities of the concurrent vector fields (in the formulation (7′)) are the same as inThomson's principle.—Someapplications of the principle to the calculation of rather sharp lower bounds are indicated, as well as an intuitivephysical interpretation (in terms of variation of masses and constraints). It is valid inthree dimensions as well; we indicate its extension toSchrodinger's equation.