Sturm-Liouville问题的特征值对势函数的依赖性

研究了定义在[0,1]上的Sturm-Liouville问题的特征值对势函数的连续依赖性.应用比较定理和定义区间单调性证明了:当部分区间[x0,1]上的势函数趋于无穷大时,[0,1]区间上的特征值渐进趋近于[0,x0]区间上的某个特征值.推广了一些作者对Sturm-Liouville问题研究的相应结果,并为其相应问题的研究提供了一个新的视角.     

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.     

The computation of negative eigenvalues of singular Sturm-Liouville problems

In this work we shall develop a new interpolation method forthe computation of eigenvalues of singular Sturm–Liouvilleproblems. Basic properties of the Jost solutions are used todetermine the growth of the boundary function and an appropriateinterpolation basis. This leads to a good approximation of thenegative eigenvalues.     

Computing eigenvalues of Sturm-Liouville problems via optimal control theory

In this paper, we shall address three problems arising in the computation of eigenvalues of Sturm-Liouville boundary value problems. We first consider a well-posed Sturm-Liouville problem with discrete and distinct spectrum. For this problem, we shall show that the eigenvalues can be computed by solving for the zeros of the boundary condition at the terminal point as a function of the eigenvalue. In the second problem, we shall consider the case where some coefficients and parameters in the differential equation are continuously adjustable. For this, the eigenvalues can be optimized with respect to these adjustable coefficients and parameters by reformulating the problem as a combined optimal control and optimal parameter selection problem. Subsequently, these optimized eigenvalues can be computed by using an existing optimal control software, MISER. The last problem extends the first to nonstandard boundary conditions such as periodic or interrelated boundary conditions. To illustrate the efficiency and the versatility of the proposed methods, several non-trivial numerical examples are included.     

Asymptotic behavior of eigenvalues of two-parameter nonlinear Sturm-Liouville problems

The nonlinear two-parameter Sturm-Liouville problemu ^"+μg(u)=λf(u) is studied for μ, λ>0. By using Ljusternik-Schnirelman theory on the general level set developed by Zeidler, we shall show the existence of ann-th variational eigenvalue λ=λ_n(μ). Furthermore, for specialf andg, the asymptotic formula of λ₁(μ)) as μ→∞ is established.     

New asymptotic formula for eigenvalues of nonlinear Sturm-Liouville problems

We study the nonlinear Sturm-Liouville problem

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.

     

Fliess series approach to the computation of the eigenvalues of fourth-order Sturm-Liouville problems

This paper is an extension to our work on the computation of the eigenvalues of regular fourth-order Sturm-Liouville problems using Fliess series. The purpose here is twofold. First, we consider general self-adjoint separated boundary conditions. Second, we modify the algorithm presented in an earlier paper to ease considerably the computation of the iterated integrals involved.     

Asymptotics of the eigenvalues of Sturm-Liouville problem

We prove a new asymptotic formula for the eigenvalues of Sturm-Liouville problem with summable potential. The obtained result extends and make more precise previously known formulas, and takes into account the smooth dependence of the spectral data on boundary conditions.     

Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

We present existence principles for the nonlocal boundary-value problem (φ(u^(p−1)))′=g(t,u,...,u^(p−1), α_k(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α _k: C ^p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)ⁿ(φ(u⁽²ⁿ⁻⁾))′=f(t,u,...,u⁽²ⁿ⁻¹⁾), u^(2k)(0)=0, α_ku^(2k)(T)+b_ku^(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.     

Estimates of eigenvalues of self-adjoint boundary-value problems with periodic coefficients

We consider self-adjoint boundary-value problems with discrete spectrum and coefficients periodic in a certain coordinate. We establish upper bounds for eigenvalues in terms of the eigenvalues of the corresponding problem with averaged coefficients. We illustrate the results obtained in the case of the Hill vector equation and for circular and rectangular plates with periodic coefficients. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 632–638, May, 1998.     

Asymptotics of eigenvalues for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition

In this paper we obtain asymptotic estimates of eigenvalues for regular Sturm-Liouville problems having the eigenparameter in the boundary condition. The method is based on an iterative procedure solving the associated Riccati equation and producing an asymptotic expansion of the solution in the higher powers of 1/λ1/2 as λ→∞.     

Accurate computation of higher Sturm-Liouville eigenvalues

Summary The computation of eigenvalues of regular Sturm-Liouville problems is considered. It is shown that a simple step-dependent linear multistep method can be used to reduce the error of the orderk ⁴ h ² of the centered finite difference estimate of thek-th eigenvalue with uniform step lengthh, to an error of orderkh ². By an appropriate minimization of the local error term of the method one can obtain even more accurate results. A comparison of the simple correction techniques of Paine, de Hoog and Anderssen and of Andrew and Paine is given. Numerical examples demonstrate the usefulness of this correction even for low values ofk.Research Director of the National Fund for Scientific Research (N.F.W.O. Belgium)     

Asymptotic behavior for differences of eigenvalues of two Sturm-Liouville problems with smooth potentials

The asymptotics for the differences of the eigenvalues of two Sturm-Liouville problems defined on [0,π] with the same boundary conditions and different smooth potentials is considered. Under the assumptions of that both problems with a suite of boundary conditions have the same one full spectrum and both potential functions and their derivatives are the same at the endpoint x=π, the asymptotic expressions associated with other boundary conditions are provided.     

Relations among eigenvalues of Sturm-Liouville problems with different types of leading coefficient functions

For any Sturm-Liouville problem with a separable boundary condition and whose leading coefficient function changes sign (exactly once), we first give a geometric characterization of its eigenvalues λn using the eigenvalues of some corresponding problems with a definite leading coefficient function. Consequences of this characterization include simple proofs of the existence of the λn's, their Prüfer angle characterization, and a way for determining their indices from the zeros of their eigenfunctions. Then, interlacing relations among the λn's and the eigenvalues of the corresponding problems are obtained. Using these relations, a simple proof of asymptotic formulas for the λn's is given.     

On the first two eigenvalues of Sturm-Liouville operators

Among the Schrödinger operators with single-well potentials defined on with transition point at , the gap between the first two eigenvalues of the Dirichlet problem is minimized when the potential is constant. This extends former results of Ashbaugh and Benguria with symmetric single-well potentials. An analogous result is given for the Dirichlet problem of vibrating strings with single-barrier densities for the ratio of the first two eigenvalues.