Discrete Sturm-Liouville Problems Research Articles

INVESTIGATION OF A DISCRETE STURM–LIOUVILLE PROBLEM WITH TWO-POINT NONLOCAL BOUNDARY CONDITION AND NATURAL APPROXIMATION OF A DERIVATIVE IN BOUNDARY CONDITION

The article investigates a discrete Sturm–Liouville problem with one natural boundary condition and another nonlocal two-point boundary condition. We analyze zeroes, poles and critical points of the characteristic function and how the properties of this function depend on parameters in nonlocal boundary condition. Properties of the Spectrum Curves are formulated and illustrated in figures.

On Some Spectral Properties of Discrete Sturm-Liouville Problem

Time scale theory helps us to combine differential equations with difference equations. Especially in models such as biology, medicine, and economics, since the independent variable is handled discrete, it requires us to analyze in discrete clusters. In these cases, the difference equations defined in $\mathbb{Z}$ are considered. Boundary value problems (BVP's) are used to solve and model problems in many physical areas. In this study, we examined spectral features of the discrete Sturm-Liouville problem. We have given some examples to make the subject understandable. The discrete Sturm-Liouville problem is solved by using the discrete Laplace transform. In the classical case, the discrete Laplace transform is preferred because it is a very useful method in differential equations and it is thought that the discrete Laplace transform will show similar properties. The other method obtained for the solution of this problem is the solutions obtained according to the states of the characteristic equation and $\lambda$ parameter. In this solution, discrete Wronskian and Cramer methods are used.

Jump phenomena of the n-th eigenvalue of discrete Sturm–Liouville problems with application to the continuous case

In this paper, we characterize jump phenomena of the $n$-th eigenvalue of self-adjoint discrete Sturm–Liouville problems in any dimension. For a fixed Sturm–Liouville equation, we completely characterize jump phenomena of the $n$-th eigenvalue. For a fixed boundary condition, unlike in the continuous case, the $n$-th eigenvalue exhibits jump phenomena and we describe the singularity under a non-degenerate assumption. Compared with the continuous case in Hu et al. (2019, J. Differ. Equ.266, 4106–4136) and Kong et al. (1999, J. Differ. Equ.156, 328–354), the jump set here is involved with coefficients of the Sturm–Liouville equations. This, along with arbitrariness of the dimension, causes difficulty when dividing the jump areas. We study the singularity by partitioning and analysing the local coordinate systems, and provide a Hermitian matrix which can determine the areas’ division. To prove the asymptotic behaviour of the $n$-th eigenvalue, we generalize the method developed in Zhu and Shi (2016, J. Differ. Equ.260, 5987–6016) to any dimension. As an application, by transforming the continuous Sturm–Liouville problem of Atkinson type to a discrete one, we determine the number of eigenvalues and obtain complete characterization of jump phenomena of the $n$-th eigenvalue for the Atkinson type.

Relations between Spectrum Curves of Discrete Sturm-Liouville Problem with Nonlocal Boundary Conditions and Graph Theory. II

In this paper, relations between discrete Sturm--Liouville problem with nonlocal integral boundary condition characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found. The previous article was devoted to the Sturm--Liouville problem in the case two-points nonlocal boundary conditions.

Reconstruction of Potential in Discrete Sturm–Liouville Problem

Reconstruction of Potential in Discrete Sturm–Liouville Problem

Global behavior and nontrivial solutions for discrete Sturm–Liouville problems with eigenparameter-dependent boundary condition

Global behavior and nontrivial solutions for discrete Sturm–Liouville problems with eigenparameter-dependent boundary condition

Relations between spectrum curves of discrete Sturm-Liouville problem with nonlocal boundary conditions and graph theory

Sturm-Liouville problem with nonlocal boundary conditions arises in many scientific fields such as chemistry, physics, or biology. There could be found some references to graph theory in a discrete Sturm-Liouville problem, especially in investigation of spectrum curves. In this paper, relations between discrete Sturm-Liouville problem with nonlocal boundary conditions characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found.

Spectrum of a Class of Difference Operators with Indefinite Weights

In this study, we use analytical methods and Sylvester inertia theorem to research a class of second order difference operators with indefinite weights and coupled boundary conditions. The eigenvalue problem with sign-changing weight has lasted a long time. The number of eigenvalues and the number of sign changes of the corresponding eigenfunctions of discrete equations under different boundary conditions are mainly studied. For the discrete Sturm-Liouville problems, similar conclusions about the properties of eigenvalues and the number of sign changes of the corresponding eigenfunctions are obtained under different boundary conditions, such as periodic boundary conditions, antiperiodic boundary conditions and separated boundary conditions etc. The purpose of this paper is to extend the similar conclusion to the coupled boundary conditions, which is of great significance to the perfection of the theory of the discrete Sturm-Liouville problems. We came to the following conclusions: first, the eigenvalues of the problem are real and single, the number of the positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. Second, under some conditions, we obtain the sign change of the eigenfunction corresponding to the j-th positive/negative eigenvalue.

Eigenvalues of discrete Sturm-Liouville problems with sign-changing weight and coupled boundary conditions

Eigenvalues of discrete Sturm-Liouville problems with sign-changing weight and coupled boundary conditions

Spectra of a discrete Sturm-Liouville problem with eigenparameter-dependent boundary conditions in Pontryagin space

In this paper, we consider the spectra of a discrete Sturm-Liouville problem with two eigenparameter-dependent boundary conditions. Different from the previous results, the operator corresponding to the problem here is not self-adjoint in the corresponding Hilbert space, but J-self-adjoint in a new Pontryagin space under a new J-metric. No matter what, the existence, simplicity and interlacing properties of real eigenvalues, nonreal eigenvalues and the oscillation properties of eigenfunctions are obtained in this paper. Meanwhile, we also obtain the existence of a pair of nonreal eigenvalues and an unconditional basis of the Pontryagin space under the new J-metric.

Spectrum curves for a discrete Sturm–Liouville problem with one integral boundary condition

This paper presents new results on the spectrum on complex plane for discrete Sturm–Liouville problem with one integral type nonlocal boundary condition depending on three parameters: γ, ξ1 and ξ2. The integral condition is approximated by the trapezoidal rule. The dependence on parameter γ is investigated by using characteristic function method and analysing spectrum curves which gives qualitative view of the spectrum for fixed ξ1 = m1 / n and ξ2 = m2 / n, where n is discretisation parameter. Some properties of the spectrum curves are formulated and illustrated in figures for various ξ1 and ξ2. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014).

A new approach for higher-order difference equations and eigenvalue problems via physical potentials

In this study, we give the variation of parameters method from a different viewpoint for the Nth order inhomogeneous linear ordinary difference equations with constant coefficient by means of delta exponential function . Advantage of this new approachment is to enable us to investigate the solution of difference equations in the closed form. Also, the method is supported with three difference eigenvalue problems, the second-order Sturm-Liouville problem, which is called also one dimensional Schr\"odinger equation, having Coulomb potential, hydrogen atom equation, and the fourth-order relaxation difference equations. We find sum representation of solution for the second order discrete Sturm-Liouville problem having Coulomb potential, hydrogen atom equation, and analytical solution of the fourth order discrete relaxation problem by the variation of parameters method via delta exponential and delta trigonometric functions .

Theory of discrete fractional Sturm–Liouville equations and visual results

In this article, we study discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Grunwald-Letnikov fractional operators with both delta and nabla operators. Self-adjointness of the DFSL operator is analyzed and fundamental spectral properties are proved. Besides, we get sum representation of solutions for DFSL problem by means of Laplace transform for nabla fractional difference equations and find the analytical solutions of the problem. Moreover, the results for DFSL problem, discrete Sturm-Liouville (DSL) problem with the second order, and fractional Sturm-Liouville (FSL) problem are compared with the second order classical Sturm-Liouville (CSL) problem. We display the results comparatively by tables and figures.

Inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems

In this paper, inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems are established. For a fixed discrete Sturm-Liouville equation, inequalities among eigenvalues for different boundary conditions are given. For a fixed boundary condition, inequalities among eigenvalues for different equations are given. These results are obtained by applying continuity and discontinuity of the n-th eigenvalue function, monotonicity in some direction of the n-th eigenvalue function, which were given in our previous papers, and natural loops in the space of boundary conditions. Some results generalize the relevant existing results about inequalities among eigenvalues of different Sturm-Liouville problems.

Direct and inverse spectral problems for discrete Sturm-Liouville problem with generalized function potential

In this work, we study the inverse problem for difference equations which are constructed by the Sturm-Liouville equations with generalized function potential from the generalized spectral function (GSF). Some formulas are given in order to obtain the matrix J, which need not be symmetric, by using the GSF and the structure of the GSF is studied.

Continuous dependence of the n-th eigenvalue of self-adjoint discrete Sturm–Liouville problems on the problem

This paper is concerned with continuous dependence of the n-th eigenvalue of self-adjoint discrete Sturm–Liouville problems on the problem. The n-th eigenvalue is considered as a function on the space of the problems, called the n-th eigenvalue function. A necessary and sufficient condition for all the n-th eigenvalue functions to be continuous and several properties of the n-th eigenvalue function on a subset of the space of the problems are given. They play an important role in the study of continuous dependence of the n-th eigenvalue function on the problem. Continuous dependence of the n-th eigenvalue function on the equation and on the boundary condition is studied separately. Consequently, the continuity and discontinuity of the n-th eigenvalue function are completely characterized on the whole space of the problems. Especially, asymptotic behavior of the n-th eigenvalue function near each discontinuity point is given.

Inverse problems for Sturm-Liouville difference equations

We consider a discrete Sturm-Liouville problem with Dirichlet boundary conditions. We show that the specification of the eigenvalues and weight numbers uniquely determines the potential. Moreover, we also show that if the potential is symmetric, then it is uniquely determined by the specification of the eigenvalues. These are discrete versions of well-known results for corresponding differential equations.

On the equivalence of discrete Sturm–Liouville problem with nonlocal boundary conditions to the algebraic eigenvalue problem

We consider the finite difference approximation of the second order Sturm–Liouville equation with nonlocal boundary conditions (NBC). We investigate the condition when the discrete Sturm–Liouville problem can be transformed to an algebraic eigenvalue problem and denote this condition as solvability condition. The examples of the solvability for the most popular NBCs are provided.
 The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/ 2014).

Dependence of Eigenvalues of Certain Closely Discrete Sturm-Liouville Problems

This paper is concerned with dependence of eigenvalues of certain closely discrete Sturm-Liouville problems. Topologies and geometric structures on various spaces of such problems are firstly introduced. Then, relationships between the analytic and geometric multiplicities of an eigenvalue are discussed. It is shown that all problems sufficiently close to a given problem have eigenvalues near each eigenvalue of the given problem. So, all the simple eigenvalues live in so-called continuous simple eigenvalue branches over the space of problems, and all the eigenvalues live in continuous eigenvalue branches over the space of self-adjoint problems. The analyticity, differentiability and monotonicity of continuous eigenvalue branches are further studied.

Closed form numerical solutions of variable coefficient linear second-order elliptic problems

In this work we develop an alternative numerical technique which allows to construct a numerical solution in closed form of variable coefficient linear second-order elliptic problems with Dirichlet boundary conditions. The elliptic partial differential equation is approximated by a consistent explicit difference scheme and using a discrete separation of the variables method we determine a closed form solution of the two resulting discrete boundary value problems with the separated variables, avoiding to have to solve large algebraic systems. One of these boundary value problems is a discrete Sturm–Liouville problem which guarantees the qualitative properties of the exact solution of elliptic problem. A constructive procedure for the computation of the numerical solution is given and an illustrative example is included.