On the eigenvalues of second-order boundary-value problems

In this paper we investigate the properties of eigenvalues of some boundary-value problems generated by second-order Sturm-Liouville equation with distributional potentials and suitable boundary conditions. Moreover, we share a necessary condition for the problem to have an infinitely many eigenvalues. Finally, we introduce some ordinary and Frechet derivatives of the eigenvalues with respect to some elements of the data.     

Asymptotic expansions of eigenvalues and eigenfunctions for elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube

We consider spectral boundary value problems of Steklov, Neumann, and Dirichlet types for second-order elliptic operators with ε-periodic coefficients in a perforated cube; the coefficients of the differential equations are assumed to satisfy some symmetry conditions. Complete asymptotic expansions with respect to the small parameter ε are constructed for eigenvalues and eigenfunctions of the said problems. Bibliography: 24 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 51–88, 1994.     

Condition of extremum for eigenvalues of elliptic boundary-value problems

In this paper, we consider problems of eigenvalue optimization for elliptic boundary-value problems. The coefficients of the higher derivatives are determined by the internal characteristics of the medium and play the role of control. The necessary conditions of the first and second order for problems of the first eigenvalue maximization are presented. In the case where the maximum is reached on a simple eigenvalue, the second-order condition is formulated as completeness condition for a system of functions in Banach space. If the maximum is reached on a double eigenvalue, the necessary condition is presented in the form of linear dependence for a system of functions. In both cases, the system is comprised of the eigenfunctions of the initial-boundary value problem. As an example, we consider the problem of maximization of the first eigenvalue of a buckling column that lies on an elastic foundation.     

Bounds on eigenvalues of Sturm-Liouville problems with discontinuous coefficients

This paper is concerned with obtaining upper and lower bounds for the eigenvalues of Sturm-Liouville problems with discontinuous coefficients. Such problems occur naturally in many areas of composite material mechanics.The problem is first transformed by using an analog of the classical Liouville transformation. Upper bounds are obtained by application of a Rayleigh-Ritz technique to the transformed problem. Explicit lower bounds in terms of the coefficients are established. Numerical examples illustrate the accuracy of the results.

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Résumé Dans cet article les bornes supérieures et inférieures sont détermineés pour les valeurs caractéristiques des problèmes de Sturm-Liouville avec des coefficients discontinus. De tels problèmes se trouvent naturellement dans la mécanique des materiaux composites.Après avoir transformé ce problème en utilisant un analogue de la transformation classique de Liouville, les bornes supérieures sont obtenues par l'application d'une technique de Rayleigh-Ritz au problème transformé. Les bornes inférieurs sont determinées en fonction des coefficients sous une forme explicite. Quelques exemples numériques montrent l'exactitude des résultats.

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This work was supported by the U.S. Army Research Office under Grants DAH C04-75-G-0059, DAAG 29-76-G-0063 and DAAG 29-77-G-0034.     

Estimates for weighted eigenvalues of a fourth-order elliptic operator with variable coefficients

We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in \mathbbRⁿ {\mathbb{R}^n} . We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the (k + 1)th eigenvalue in terms of the first k eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang–Xia inequality (J. Funct. Anal., 245, No. 1, 334–352 (2007)) for the clamped-plate problem to a fourth-order elliptic operator with variable coefficients.     

Operational methods and solutions of boundary-value problems with periodic data

We introduce suitable exponential operators and use the multi-dimensional polynomials considered by Hermite and subsequently studied by P. Appell and J. Kampé de Fériet, H. W. Gould and A. T. Hopper, and G. Dattoli et al. in order to obtain explicit solutions of classical boundary-value problems with periodic data. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

High order approximations of the eigenvalues of sturm-liouville problems with coupled self-adjoint boundary conditions

Sampling theory has been used to compute with great accuracy the eigenvalues of regular and singular Sturm-Liouville problems of Bessel Type. We shall consider in this paper the case of general coupled real or complex self-adjoint boundary conditions. We shall present few examples to illustrate the power of the method and compare our results with the ones obtained using the well-known Sleign2 package.     

An approach for computing eigenvalues of discrete symplectic boundary-value problems

We propose a new method for calculating eigenvalues of discrete symplectic boundaryvalue problems. This method is based on the discrete oscillation theory and on a modification of the Abramov double sweep method for discrete self-adjoint boundary-value problems.     

Estimation of eigenvalues of self-adjoint operators

Estimates of the number of eigenvalues are obtained for perturbations of certain self-adjoint and unitary operators in a Hilbert space. In particular, we consider a perturbation of the operator of multiplication by an independent variable inL ₂ () andL ₂ (0, 1).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 642–648, May, 1994.     

Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

This is a brief survey of M. G. Krein's contribution to the theory of self-adjoint extensions of Hermitian operators and to the theory of boundary-value problems for differential equations. The further development of these results is also considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 55–62, January–February, 1994.     

An approximate solution for linear boundary-value problems with slowly varying coefficients

In this paper, an approximate closed-form solution for linear boundary-value problems with slowly varying coefficient matrices is obtained. The derivation of the approximate solution is based on the freezing technique, which is commonly used in analyzing the stability of slowly varying initial-value problems as well as solving them. The error between the approximate and the exact solutions is given, and an upper bound on the norm of the error is obtained. This upper bound is proportional to the rate of change of the coefficient matrix of the boundary-value problem. The proposed approximate solution is obtained for a two-point boundary-value problem and is compared to its solution obtained numerically. Good agreement is observed between the approximate and the numerical solutions, when the rate of change of the coefficient matrix is small.     

Global bifurcation of a class of periodic boundary-value problems

We prove that λ=0 is a global bifurcation point of the second-order periodic boundary-value problem (p(t)x^′(t))^′−λx(t)−λ²x^′(t)−f(t,x(t),x^′(t),x^″(t));x(0)=x(1),x^′(0)=x^′(1). We study this equation under hypotheses for which it may be solved explicitly for x^″(t). However, it is shown that the explicitly solved equation does not satisfy the usual conditions that are sufficient to conclude global bifurcation. Thus, we need to study the implicit equation with regard to global bifurcation.     

Some remarks on bounds to eigenvalues of Sturm-Liouville problems with discontinuous coefficients

Upper and lower bounds on the eigenvalues of Sturm-Liouville problems with discontinuous coefficients are discussed. Rayleigh-Ritz approximations based both on Rayleigh's quotient and the dual Rayleigh quotient are used for obtaining upper bounds for the eigenvalues. Though previous studies have indicated that such approximations yield poor results when large discontinuities in the coefficients occur, it is shown in this paper by means of numerical examples that thesame rate of convergence can be achieved as for systems with continuous coefficients, provided the trial functions are allowed to have arbitrary jump discontinuities in their derivatives across the points where the coefficients suffer discontinuities. New explicit lower bounds in terms of the coefficients are also established. The accuracy of the new estimates is illustrated by numerical examples.

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Résumé On discute les bornes supérieures et inférieures des valeurs caractéristiques des problèmes de Sturm-Liouville avec des coefficients discontinus. Les approximations de Rayleigh-Ritz, basées sur le quotient de Rayleigh et le quotient jumelé de Rayleigh, sont utilisées pour obtenir les bornes supérieures des valeurs caractéristiques. Bien que les études antérieures aient indiqué que ces approximations donnent des résultats médiocres quand les coefficients ont de grandes discontinuités, on démontre dans cet article par des exemples numériques qu'on peut réaliser le même degré de convergence que pour les systèmes á coefficients continuous, pourvu que les fonctions d'essai admises aient des sauts arbitraires dans leurs dérivées á travers les points où les coefficients subissent des discontinuités. De nouvelles bornes inférieures sont déterminées sous une forme explicite en fonction des coefficients. On montre l'exactitude des nouveaux résultats par des exemples numériques.

     

Estimates for the inverse of tridiagonal matrices arising in boundary-value problems

By considering tridiagonal matrices as three-term recurrence relations with Dirichlet boundary conditions, one can formulate their inverses in terms of Green's functions. This analysis is applied to three-point difference schemes for 1-D problems, and five-point difference schemes for 2-D problems. We derive either an explicit inverse of the Jacobian or a sharp estimate for both uniform and nonuniform grids.