A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

2014 ◽  
Vol 19 (3) ◽  
pp. 301-334 ◽  
Author(s):  
Artūras Štikonas ◽  
Keyword(s):  
Boundary Conditions ◽  
Green's Functions ◽  
Green’S Functions ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Sturm Liouville ◽  
Stationary Problems

scholarly journals CHARACTERISTIC FUNCTIONS FOR STURM—LIOUVILLE PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

2009 ◽  
Vol 14 (2) ◽  
pp. 229-246 ◽  
Author(s):  
Artūras Štikonas ◽  
Olga Štikonienė
Keyword(s):  
Characteristic Function ◽  
Boundary Conditions ◽  
Function Method ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Characteristic Functions ◽  
Sturm Liouville ◽  
Definition Of

This paper presents some new results on a spectrum in a complex plane for the second order stationary differential equation with one Bitsadze‐Samarskii type nonlocal boundary condition. In this paper, we survey the characteristic function method for investigation of the spectrum of this problem. Some new results on characteristic functions are proved. Many results of this investigation are presented as graphs of characteristic functions. A definition of constant eigenvalues and the characteristic function is introduced for the Sturm‐Liouville problem with general nonlocal boundary conditions.

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

2021 ◽  
Vol 62 ◽  
pp. 1-8
Author(s):  
Jonas Vitkauskas ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Graph Theory ◽  
Boundary Conditions ◽  
Critical Points ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Integral Boundary Condition ◽  
Integral Boundary ◽  
Sturm Liouville

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.

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

2021 ◽  
Vol 61 ◽  
pp. 1-6
Author(s):  
Jonas Vitkauskas ◽  
Artūras Štikonas
Keyword(s):  
Graph Theory ◽  
Boundary Conditions ◽  
Critical Points ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Sturm Liouville ◽  
Scientific Fields

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.

scholarly journals INVESTIGATION OF SPECTRUM CURVES FOR A STURM-LIOUVILLE PROBLEM WITH TWO-POINT NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
Vol 25 (1) ◽  
pp. 37-52
Author(s):  
Kristina Bingelė ◽  
Agnė Bankauskienė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Characteristic Function ◽  
Boundary Conditions ◽  
Critical Points ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Point Boundary ◽  
Sturm Liouville

The article investigates the Sturm–Liouville problem with one classical 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 for various values of parameter ξ.

INVESTIGATION OF CHARACTERISTIC CURVE FOR STURM–LIOUVILLE PROBLEM WITH NONLOCAL BOUNDARY CONDITIONS ON TORUS

2011 ◽  
Vol 16 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Artūras Štikonas
Keyword(s):  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Characteristic Curve ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Zero Eigenvalue ◽  
Extended Plane ◽  
Sturm Liouville ◽  
Two Parameters

In this paper, we investigate the second-order Sturm–Liouville problem with two additional Nonlocal Boundary Conditions. Nonlocal boundary conditions depends on two parameters. We find condition for existence of zero eigenvalue in the parameters space and classified Characteristic Curves in the plane and extended plane is described as torus. The Characteristic Curve on torus may be of three types only. Some new conclusions about existence and uniqueness domain of solution are presented.

scholarly journals Linear ODE with nonlocal boundary conditions and Green’s functions for such problems

2019 ◽  
Vol 51 ◽  
pp. 379-384
Author(s):  
Svetlana Roman ◽  
Artūras Štikonas
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Linear Ode

In this article we investigate a formula for the Green’s function for the n-orderlinear differential equation with n additional conditions. We use this formula for calculatingthe Green’s function for problems with nonlocal boundary conditions.

scholarly journals The properties of Green’s functions for one stationary problem with nonlocal boundary conditions

2019 ◽  
Vol 50 ◽  
Author(s):  
Svetlana Roman ◽  
Artūras Štikonas
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Stationary Problem

In this paper we research Green’s function properties for stationary problem with four-pointnonlocal boundary conditions. Dependence of these functions on values ξ and γ is investigated. Green’sfunctions graphs with various values ξ and γ are presented.

scholarly journals Generalized Green’s functions for second-order discrete boundary-value problems with nonlocal boundary conditions

2012 ◽  
Vol 53 ◽  
pp. 96-101 ◽  
Author(s):  
Gailė Paukštaitė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Conditions ◽  
Green's Functions ◽  
Green’S Functions ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence Condition ◽  
Second Order ◽  
Linear Functionals ◽  
Discrete Green’S Function ◽  
Sufficient Existence Condition

In this paper, generalized Green’s functions for second-order discrete boundaryvalueproblems with nonlocal boundary conditions are investigated, where the necessaryand sufficient existence condition of discrete Green’s function is not satisfied and nonlocalboundary conditions are described by linear functionals.

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

2015 ◽  
Vol 56 ◽  
pp. 66-71
Author(s):  
Jurij Novickij ◽  
Artūras Štikonas
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Solvability Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Sturm Liouville ◽  
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).

Sign in / Sign up

Export Citation Format

Share Document