We study finite difference approximations of solutions of direct and inverse Sturm–Liouville problems, in a finite or infinite interval on the real line. The discretization is done on optimal grids, with a three-point finite difference stencil. The optimal location of the grid points is calculated via a rational approximation of the Neumann-to-Dirichletmap and the latter converges exponentially...

In this article we have discovered a close relationship between the (algebraic) Bethe Ansatz equations of the spin s XXZ model of a finite size and the q-Sturm-Liouville problem. We have demonstrated that solutions of the Bethe Ansatz equations give rise to the polynomial solutions of a second order q-difference equation in terms of Askey-Wilson operator. The more general form of Bethe Ansatz e...

for some λ ∈ C, x ∈ I = [a, b], and y ∈ C2(I). It was first introduced in a 1837 publication [7] by the eminent French mathematicians Joseph Liouville (1809 1882) and Jacques Charles François Sturm (1803 1855). At this point, our initial questions might be: why is this problem important? What can we say about the structure of this equation? How about the solutions? We will tackle a few basic re...

*Correspondence: [email protected] Department of Mathematics, Gaziosmanpasa University, Tasliciftlik Campus, Tokat, 60250, Turkey Abstract The main purpose of this study is to investigate a fractional discontinuous Sturm-Liouville problem with transmission conditions. We shall consider a fractional boundary value problem involving an operator with two parts. It is shown that the eigen...

Consider the minimal Sturm-Liouville operator A = Amin generated by the differential expression

We study the nonlinear boundary value problem consisting of the equation y+w(t)f(y) = 0 on [a, b] and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a twopoint separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes of the existence of different...

In this paper, we derive Haar wavelet operational matrix of the fractional integration and use it to obtain eigenvalues of fractional Sturm-Liouville problem. The fractional derivative is described in the Caputo sense. The efficiency of the method is demo nstrated by examples MSC: 26A33

In this paper, we derive Haar wavelet operational matrix of the fractional integration and use it to obtain eigenvalues of fractional Sturm-Liouville problem. The fractional derivative is described in the Caputo sense. The efficiency of the method is demonstrated by examples MSC: 26A33

For any positive integer n and any given n distinct real numbers we construct a Sturm-Liouville problem whose spectrum is precisely the given set of n numbers. Such problems are of Atkinson type in the sense that the weight function or the reciprocal of the leading coefficient is identically zero on at least one subinterval.

In this paper we present a new method for proving the existence of an eigenvalue below the essential spectrum for Sturm-Liouville operators with coefficients which are L1 perturbations of periodic functions. This method combines operator theory and “standard” numerical analysis with interval arithmetic analysis. We illustrate the method by showing that the SturmLiouville problem (SLP) consistin...