Singular Sturm-Liouville Problems Research Articles

Spectra of Nonselfadjoint Eigenvalue Problems for Elliptic Systems in Mathematical Models of the Wave Propagation in Open Waveguides

Statements and analysis are presented of nonselfadjoint eigenvalue problems for elliptic equations and systems, including singular Sturm–Liouville problems on the line, that arise in mathematical models of the wave propagation in open metal-dielectric waveguides. Existence of real and complex spectra are proved and their distribution is investigated for canonical structures possessing circular symmetry of boundary contours.

The inverse spectral problem of some singular Sturm–Liouville problems with sign-valued weights

In this paper we study the inverse spectral problem for singular Sturm–Liouville operator with sign valued weight. We define the spectral data of the problem, we construct the main integral equation. Solve the inverse spectral problem and use the inverse problem by two spectrum.

A note on the quasi-stationary distribution of the Shiryaev martingale on the positive half-line

В работе получена явная формула для квазистационарного распределения классического мартингального диффузионного процесса Ширяева, рассматриваемого на луче $[A,+\infty)$, где $A>0$ фиксировано; предполагается, что левая граница фазового пространства процесса является поглощающей. Вывод формулы осуществляется аналитически, посредством решения соответствующей вырожденной задачи Штурма-Лиувилля; последняя была впервые рассмотрена в [7, п. 7.8.2], однако оставалась нерешенной до настоящего момента.

Weyl Function of Sturm–Liouville Problems with Transmission Conditions at Finite Interior Points

In this paper, a class of singular Sturm–Liouville problems with transmission conditions at finite interior points is studied. The equation, center and radius of Weyl circle are given. The necessary and sufficient conditions for the function $$ m(\lambda ) $$ , which is in the Weyl circle, are established. The definition of Weyl function for singular Sturm–Liouville problems with transmission conditions in the case of limit circle is given, and the properties of Weyl function are discussed.

ON THE SPECTRAL STUDY OF SINGULAR STURM-LIOUVILLE PROBLEM WITH SIGN-VALUED WEIGHT Vol. 5(2) July 2017, No. 11, pp. 98-115 ZAKI F.A. EL-RAHEEM AND SHIMAA A.M. HAGAG

ON THE SPECTRAL STUDY OF SINGULAR STURM-LIOUVILLE PROBLEM WITH SIGN-VALUED WEIGHT Vol. 5(2) July 2017, No. 11, pp. 98-115 ZAKI F.A. EL-RAHEEM AND SHIMAA A.M. HAGAG

A Singular Sturm-Liouville Problem with Limit Circle Endpoints and Eigenparameter Dependent Boundary Conditions

In this paper, we investigate a class of discontinuous singular Sturm-Liouville problems with limit circle endpoints and eigenparameter dependent boundary conditions. Operator formulation is constructed and asymptotic formulas for eigenvalues and fundamental solutions are given. Moreover, the completeness of eigenfunctions is discussed.

Computing eigenelements of Sturm-Liouville problems by using Daubechies wavelets

This work is our first step to get multiresolution approximation of eigenelements of Sturm-Liouville problems within bounded domain of varied nature. The formula for obtaining elements of representation of Sturm-Liouville operator involving polynomial coefficients in wavelet basis of Daubechies family have been derived in a form which can be readily used for their computations by a simple computer program. Estimates of errors for both the eigenvalues and eigenfunctions are also presented here. The proposed wavelet-Galerkin scheme based on scale functions and wavelets of Daubechies family having three or four vanishing moments of their wavelets has been applied to get approximate eigenelements of regular and singular Sturm-Liouville problems within bounded domain and compared with the exact or approximate results whenever available. From our study it appears that the proposed method is efficient and rapidly convergent in comparison to other approximation schemes based on variational method in Haar basis or finite difference methods studied by Bujurke et al. [39].

Computation of Some Second Order Sturm-Liouville Bvps using Chebyshev-Legendre Collocation Method

We propose Chebyshev-Legendre spectral collocation method for solving second order linear and nonlinear eigenvalue problems exploiting Legendre derivative matrix. The Sturm-Liouville (SLP) problems are formulated utilizing Chebyshev-Gauss-Lobatto (CGL) nodes instead of Legendre Gauss-Lobatto (LGL) nodes and Legendre polynomials are taken as basis function. We discuss, in details, the formulations of the present method for the Sturm-Liouville problems (SLP) with Dirichlet and mixed type boundary conditions. The accuracy of this method is demonstrated by computing eigenvalues of three regular and two singular SLP's. Nonlinear Bratu type problem is also tested in this article. The numerical results are in good agreement with the other available relevant studies.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 95-112

Matslise 2.0

The Matslise 2.0 software package is a thorough revision of the successful Matlab package Matslise of 2005. The package can be used to compute the eigenvalues and eigenfunctions of regular and some important classes of singular self-adjoint Sturm-Liouville boundary value problems. The code uses new or improved algorithms, offers some new features, and has an updated graphical user interface.

The double exponential sinc collocation method for singular Sturm-Liouville problems

Sturm-Liouville problems are abundant in the numerical treatment of scientific and engineering problems. In the present contribution, we present an efficient and highly accurate method for computing eigenvalues of singular Sturm-Liouville boundary value problems. The proposed method uses the double exponential formula coupled with sinc collocation method. This method produces a symmetric positive-definite generalized eigenvalue system and has exponential convergence rate. Numerical examples are presented and comparisons with single exponential sinc collocation method clearly illustrate the advantage of using the double exponential formula.

Eigenvalue estimates for operators with finitely many negative squares

Let \(A\) and \(B\) be selfadjoint operators in a Krein space. Assume that the resolvent difference of \(A\) and \(B\) is of rank one and that the spectrum of \(A\) consists in some interval \(I\subset\mathbb{R}\) of isolated eigenvalues only. In the case that \(A\) is an operator with finitely many negative squares we prove sharp estimates on the number of eigenvalues of \(B\) in the interval \(I\). The general results are applied to singular indefinite Sturm-Liouville problems.

Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Recently, we used the Sinc collocation method with the double exponential transformation to compute eigenvalues for singular Sturm-Liouville problems. In this work, we show that the computation complexity of the eigenvalues of such a differential eigenvalue problem can be considerably reduced when its operator commutes with the parity operator. In this case, the matrices resulting from the Sinc collocation method are centrosymmetric. Utilizing well known properties of centrosymmetric matrices, we transform the problem of solving one large eigensystem into solving two smaller eigensystems. We show that only 1/(N+1) of all components need to be computed and stored in order to obtain all eigenvalues, where (2N+1) corresponds to the dimension of the eigensystem. We applied our result to the Schr\"odinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.

Reprint of Variable-step finite difference schemes for the solution of Sturm–Liouville problems

We discuss the solution of regular and singular Sturm–Liouville problems by means of High Order Finite Difference Schemes. We describe a method to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are considered to emphasize the behavior of a code based on the proposed algorithm. The methods solve any regular or singular Sturm–Liouville problem, providing high accuracy and computational efficiency thanks to the powerful strategy of stepsize variation.

Regular approximation of singular Sturm–Liouville problems with transmission conditions

In this paper we study regular approximation of singular Sturm–Liouville problems with transmission conditions. We construct all self-adjoint extensions of the minimal operators and the induced restriction operators. Using the abstract operator theory in the Hilbert space, we obtain that the spectrum of singular Sturm–Liouville problems with transmission conditions can be approximated by eigenvalues of a sequence of regular Sturm–Liouville problems with transmission conditions.

Variable-step finite difference schemes for the solution of Sturm–Liouville problems

We discuss the solution of regular and singular Sturm–Liouville problems by means of High Order Finite Difference Schemes. We describe a method to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are considered to emphasize the behavior of a code based on the proposed algorithm. The methods solve any regular or singular Sturm–Liouville problem, providing high accuracy and computational efficiency thanks to the powerful strategy of stepsize variation.

Existence and Uniqueness of Positive Solutions for Fourth-Order Nonlinear Singular Sturm-Liouville Problems

By mixed monotone method, we establish the existence and uniqueness of positive solutions for fourth-order nonlinear singular Sturm-Liouville problems. The theorems obtained are very general and complement previously known results.

Computation of topological degree in ordered Banach spaces with lattice structure and applications

Using the cone theory and the lattice structure, we establish some methods of computation of the topological degree for the nonlinear operators which are not assumed to be cone mappings. As applications, existence results of nontrivial solutions for singular Sturm-Liouville problems are given. The nonlinearity in the equations can take negative values and may be unbounded from below.

Fractional singular Sturm-Liouville operator for Coulomb potential

In this paper, we define a fractional singular Sturm-Liouville operator having Coulomb potential of type . Our main issue is to investigate the spectral properties for the operator. Furthermore, we prove new results according to the fractional singular Sturm-Liouville problem. MSC:26A33, 34A08.

Reconstruction of potential function for Sturm-Liouville operator with Coulomb potential

In this paper, we are concerned with an inverse problem for the Sturm-Liouville operator with Coulomb potential using a new kind of spectral data that is known as nodal points. We give a reconstruction of q as a limit of a sequence of functions whose n th term is dependent only on eigenvalue and its associated nodal data. It is mentioned that this method is based on the works of Law and Yang, but we have applied the method to the singular Sturm-Liouville problem. MSC:34L05, 45C05.

Positive solutions to a singular second order boundary value problem

In this paper, we investigate the existence and multiplicity of positive solutions for a singular second order scalar Sturm-Liouville boundary value problem with different values of λ for a function f involve u by applying the Krasnosel’skii fixed point theorem on compression and expansion of cones.