scholarly journals Scattering theory of impulsive Sturm-Liouville equations

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5401-5409 ◽  
Author(s):  
Elgiz Bairamov ◽  
Yelda Aygar ◽  
Basak Eren
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Scattering Theory ◽  
Boundary Value ◽  
Scattering Function ◽  
Jost Function ◽  
Impulsive Boundary Value Problem ◽  
Sturm Liouville ◽  
Jost Solution

In this paper, we investigate scattering theory of the impulsive Sturm-Liouville boundary value problem (ISBVP). In particular, we find the Jost solution and the scattering function of this problem. We also study the properties of the Jost function and the scattering function of this ISBVP. Furthermore, we present two examples by getting Jost function and scattering function of the impulsive boundary value problem. Besides, we examine the eigenvalues of these boundary value problems given in examples in detail.

scholarly journals Multiplicity of positive solutions for second order Sturm-Liouville boundary value problems

2016 ◽  
Vol 25 (2) ◽  
pp. 215-222
Author(s):  
K. R. PRASAD ◽  
◽  
N. SREEDHAR ◽  
L. T. WESEN ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Multiple Positive Solutions ◽  
Sturm Liouville ◽  
Multiplicity Of Positive Solutions

In this paper, we develop criteria for the existence of multiple positive solutions for second order Sturm-Liouville boundary value problem, u 00 + k 2u + f(t, u) = 0, 0 ≤ t ≤ 1, au(0) − bu0 (0) = 0 and cu(1) + du0 (1) = 0, where k ∈ 0, π 2 is a constant, by an application of Avery–Henderson fixed point theorem.

A Discreate Calculus with Applications of High-Order Discretizations to Boundary-Value Problems

2004 ◽  
Vol 4 (2) ◽  
pp. 228-261 ◽  
Author(s):  
Stanly Steinberg
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Energy Inequality ◽  
Numerical Test ◽  
Mixed Boundary Conditions ◽  
Discrete Analog ◽  
High Order ◽  
Sturm Liouville

AbstractWe develop a discrete analog of the differential calculus and use this to develop arbitrarily high-order approximations to Sturm–Liouville boundary-value problems with general mixed boundary conditions. An important feature of the method is that we obtain a discrete exact analog of the energy inequality for the continuum boundary-value problem. As a consequence, the discrete and continuum problems have exactly the same solvability conditions. We call such discretizations mimetic. Numerical test confirm the accuracy of the discretization. We prove the solvability and convergence for the discrete boundary-value problem modulo the invertibility of a matrix that appears in the discretization being positive definite. Numerical experiments indicate that the spectrum of this matrix is real, greater than one, and bounded above by a number smaller than three.

scholarly journals Discrete Sturm-Liouville equations with point interaction

2022 ◽  
Author(s):  
Güher Özbey ◽  
yelda AYGAR ◽  
Basak Oznur
Keyword(s):  
Boundary Value Problem ◽  
Discrete Spectrum ◽  
Resolvent Operator ◽  
Boundary Value ◽  
Point Interaction ◽  
Scattering Function ◽  
Asymptotic Equation ◽  
Sturm Liouville

Scattering solutions and several properties of scattering function of a discrete Sturm-Liouville boundary value problem with point interaction (PBVP) are derived. Moreover, resolvent operator, continuous and discrete spectrum of this PBVP are investigated. An asymptotic equation is utilized to get the properties of eigenvalues. An example illustrating the main results is given.

scholarly journals Note on the conformable boundary value problems: Sturm’s theorems and Green’s function

2021 ◽  
Vol 67 (3 May-Jun) ◽  
pp. 471
Author(s):  
F. Martínez ◽  
I. Martínez ◽  
M. K. A. Kaabar ◽  
S. Paredes
Keyword(s):  
Boundary Value Problem ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Boundary Value ◽  
Liouville Problem ◽  
Comparison Theorems ◽  
Conformable Derivative ◽  
Sturm Liouville ◽  
Linear Differential

Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.

scholarly journals TRANSFORMATION OF STURM–LIOUVILLE PROBLEMS WITH DECREASING AFFINE BOUNDARY CONDITIONS

2004 ◽  
Vol 47 (3) ◽  
pp. 533-552 ◽  
Author(s):  
Paul A. Binding ◽  
Patrick J. Browne ◽  
Warren J. Code ◽  
Bruce A. Watson
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Subject Classification ◽  
Darboux Transformations ◽  
Sturm Liouville

AbstractWe consider Sturm–Liouville boundary-value problems on the interval $[0,1]$ of the form $-y''+qy=\lambda y$ with boundary conditions $y'(0)\sin\alpha=y(0)\cos\alpha$ and $y'(1)=(a\lambda+b)y(1)$, where $a\lt0$. We show that via multiple Crum–Darboux transformations, this boundary-value problem can be transformed ‘almost’ isospectrally to a boundary-value problem of the same form, but with the boundary condition at $x=1$ replaced by $y'(1)\sin\beta=y(1)\cos\beta$, for some $\beta$.AMS 2000 Mathematics subject classification: Primary 34B07; 47E05; 34L05

scholarly journals Scattering analysis and spectrum of discrete Schrödinger equations with transmission conditions

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5391-5399 ◽  
Author(s):  
Elgiz Bairamov ◽  
Yelda Aygar ◽  
Dilara Karslıoğlu
Keyword(s):  
Boundary Value Problem ◽  
Discrete Spectrum ◽  
Boundary Value ◽  
Transmission Conditions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Scattering Function ◽  
Dinger Equation ◽  
Special Case ◽  
Jost Solution

In this paper, we present an investigation about scattering analysis of an transmission boundary value problem (TBVP) which consists a discrete Schr?dinger equation and transmission conditions. Discussing the Jost solution and scattering function of this problem, we find the properties of scattering function of this problem by using the scattering solutions. We also investigate the discrete spectrum of this boundary value problem. Furthermore, we apply the results on an example which is the special case of main TBVP and we discuss the existence of eigenvalues of this example.

scholarly journals The Levinson-Type Formula for a class of Sturm-Liouville Equation

2021 ◽  
pp. 1219
Author(s):  
Sertac Goktas ◽  
Khanlar R. Mamedov
Keyword(s):  
Boundary Value Problem ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Boundary Value ◽  
Scattering Data ◽  
Scattering Function ◽  
Type Formula ◽  
Marchenko Equation ◽  
Sturm Liouville

The boundary value problem \[-{\psi}''+q(x)\psi={\lambda}^2 \psi, \quad 0<x<\infty,\] \[{\psi}'(0)-(\alpha_{0}+\alpha_{1}\lambda){\psi}(0)=0 \] is considered, where $\lambda$ is a spectral parameter, $ q(x) $ is real-valued function such that \begin{equation*} \int\limits_{0}^{\infty}(1+x)|q(x)|dx<\infty \end{equation*} with $\alpha_{0}, \alpha_{1}\geq0$ ( $\alpha_{0},\alpha_{1}\in \mathbb{R}$). In this paper, for the above-mentioned boundary value problem, the scattering data is considered and the characteristics properties (such as continuity of the scattering function $ S(\lambda) $ and giving the Levinson-type formula) of this data are studied.{\small \bf Keywords. }{Scattering data, scattering function, Gelfand-Levitan-Marchenko equation, Levinson-type formula.}

scholarly journals Existence principles for second order nonresonant boundary value problems

1994 ◽  
Vol 7 (4) ◽  
pp. 487-507 ◽  
Author(s):  
Donal O'Regan
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Second Order ◽  
Carathéodory Function ◽  
Sturm Liouville

We discuss the two point singular “nonresonant” boundary value problem 1p(py′)′=f(t,y,py′) a.e. on [0,1] with y satisfying Sturm Liouville, Neumann, Periodic or Bohr boundary conditions. Here f is an L1-Carathéodory function and p∈C[0,1]∩C1(0,1) with p>0 on (0,1).

scholarly journals TYPES AND MULTIPLICITY OF SOLUTIONS TO STURM–LIOUVILLE BOUNDARY VALUE PROBLEM

2015 ◽  
Vol 20 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Maria Dobkevich ◽  
Felix Sadyrbaev
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Second Order ◽  
Nonlinear Boundary Value Problems ◽  
Multiplicity Of Solutions ◽  
Different Types ◽  
Sturm Liouville

We consider the second-order nonlinear boundary value problems (BVPs) with Sturm–Liouville boundary conditions. We define types of solutions and show that if there exist solutions of different types then there exist intermediate solutions also.

An inverse problem for Sturm–Liouville operators with non-separated boundary conditions containing the spectral parameter

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Chinare G. Ibadzadeh ◽  
Ibrahim M. Nabiev
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Spectral Data ◽  
Spectral Parameter ◽  
Boundary Value ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Liouville Operators

AbstractIn this paper a boundary value problem is considered generated by the Sturm–Liouville equation and non-separated boundary conditions, one of which contains a spectral parameter. We give a uniqueness theorem, develop an algorithm for solving the inverse problem of reconstruction of boundary value problems with spectral data. We use the spectra of two boundary value problems and some sequence of signs as a spectral data.

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