Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem

Abstract

We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piecewise continuous potential together with eigenparameter dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular, it is shown that the problem under consideration has precisely denumerable many eigenvalues λ 1 , λ 2 , ..., which are real and tends to +∞. Moreover, it is proven that the generalized eigenvectors form a Riesz basis of the adequate Hilbert space.

A new finite difference algorithm for boundary value problems involving transmission conditions

“…At first we consider the problem ( 12)-( 14) without transmission conditions (14). It is easy to verify that the function…”

“…One of them is the finite difference method (FDM), which can be applied to wide class of problems appearing in mathematical physics and engineering. Many important theoretical and numerical results have been obtained during the last seven decades regarding the stability, accuracy and convergence of the FDM for different type initial and/or boundary value problems (see, [1,2,5,6,14] and references cited therein). The standard FDM is intended for solving one-interval initial and/or boundary value problems without transmission conditions (see, [7,8,15,16]).…”

Two‐linked periodic Sturm–Liouville problems with transmission conditions

“…In a previous study, 13 a new technique is developed to estimate the eigenvalues of non-selfadjoint Sturm-Liouville problems with periodic and anti-periodic boundary conditions. In recent years, there has been growing interest of Sturm-Liouville problems with additional transmission conditions at some interior points of interaction, because of the appearance of new and interesting applications in mathematical physics (see, e.g., previous studies [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] and the references therein). For example, in electrostatics and magnetostatics, the model problem which describes the heat transfer through an infinitely conductive layer is a transmission problem.…”

“…In recent years, there has been growing interest of Sturm–Liouville problems with additional transmission conditions at some interior points of interaction, because of the appearance of new and interesting applications in mathematical physics (see, e.g., previous studies 14–28 and the references therein). For example, in electrostatics and magnetostatics, the model problem which describes the heat transfer through an infinitely conductive layer is a transmission problem 29 .…”

Modified Finite Difference Method for Solution of Two-interval Boundary Value Problems with Transition Conditions

“…Based on FDM, we have developed a new technique for solving two-interval Sturm-Liouville problems (SLPs), that included additional transition conditions across the common endpoint of these intervals. We note that, some important theoretical aspects of Sturm-Liouville problems with transition conditions were studied in [7,10,11].…”