scholarly journals Asymptotic analysis of Sturm–Liouville problem with nonlocal integral-type boundary condition

2021 ◽  
Vol 26 (5) ◽  
pp. 969-991
Author(s):  
Artūras Štikonas ◽  
Erdoğan Şen
Keyword(s):  
Boundary Condition ◽  
Arbitrary Order ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Asymptotic Formulas ◽  
Classical Type ◽  
Integral Type ◽  
Sturm Liouville ◽  
Type Boundary

In this study, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the one-dimensional Sturm–Liouville equation with one classical-type Dirichlet boundary condition and integral-type nonlocal boundary condition. We investigate solutions of special initial value problem and find asymptotic formulas of arbitrary order. We analyze the characteristic equation of the boundary value problem for eigenvalues and derive asymptotic formulas of arbitrary order. We apply the obtained results to the problem with integral-type nonlocal boundary condition.

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

2019 ◽  
Vol 24 (5) ◽  
Author(s):  
Kristina Bingelė ◽  
Agnė Bankauskienė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Function Method ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Integral Condition ◽  
Trapezoidal Rule ◽  
Integral Type ◽  
Integral Boundary ◽  
Sturm Liouville

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).

scholarly journals Investigation of the spectrum for Sturm–Liouville problems with a nonlocal boundary condition

2013 ◽  
Vol 54 ◽  
pp. 73-78
Author(s):  
Kristina Skučaitė-Bingelė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Characteristic Function ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
The Other ◽  
Sturm Liouville ◽  
Type Boundary ◽  
Type Boundary Condition

In this paper, we analyze the Sturm–Liouville problem with one classical first type boundary condition and the other Samarskii–Bitsadze type nonlocal boundary condition. We investigate how the spectrum of this problem depends on the parameters γ and ξ  of the nonlocal boundary condition. Some new results are given as graphs of the characteristic function.

scholarly journals Investigation of the spectrum of the Sturm–Liouville problem with a nonlocal integral condition

2013 ◽  
Vol 54 ◽  
pp. 67-72
Author(s):  
Agnė Skučaitė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Integral Condition ◽  
Characteristic Functions ◽  
Integral Type ◽  
Complex Characteristic ◽  
Nonlocal Integral Condition ◽  
Sturm Liouville

This paper presents some new results on the spectrum for the second order dif-ferential problem with one integral type nonlocal boundary condition (NBC). We investigate how the spectrum of this problem depends on the integral nonlocal boundary condition pa-rameters γ, ξ and the symmetric interval in the integral. Some new results are given on the complex spectra of this problem. Many results are presented as graphs of real and complex characteristic functions.

scholarly journals Zeroes and poles of a characteristic function for Sturm–Liouville problem with nonlocal integral condition

2015 ◽  
Vol 56 ◽  
pp. 95-100
Author(s):  
Agnė Skučaitė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Research Council ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Integral Condition ◽  
Integral Type ◽  
Nonlocal Integral Condition ◽  
Sturm Liouville ◽  
Special Points

This paper presents some new results on a spectrum for the Sturm–Liouville problem with one integral type nonlocal boundary condition depending on tree parameters (γ, ξ1, ξ2). Some new results on distribution of the sets of special points (poles, zeros and constant eigenvalue points) are presented. The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/ 2014).

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

scholarly journals Investigation of Negative Critical Points of the Characteristic Function for Problems with Nonlocal Boundary Conditions

2008 ◽  
Vol 13 (4) ◽  
pp. 467-490 ◽  
Author(s):  
S. Pečiulytė ◽  
O. Štikonienė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Characteristic Function ◽  
Critical Points ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
The Other ◽  
Integral Boundary ◽  
Sturm Liouville

In this paper the Sturm-Liouville problem with one classical and the other nonlocal two-point or integral boundary condition is investigated. Critical points of the characteristic function are analyzed. We investigate how distribution of the critical points depends on nonlocal boundary condition parameters. In the first part of this paper we investigate the case of negative critical points.

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.

scholarly journals Investigation of complex eigenvalues for a stationary problem with two-point nonlocal boundary condition

2011 ◽  
Vol 52 ◽  
pp. 303-308
Author(s):  
Kristina Skučaitė-Bingelė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Finite Difference ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Stationary Problem ◽  
Finite Difference Schemes ◽  
Complex Eigenvalues ◽  
Sturm Liouville

The Sturm–Liouville problem with one classical and another two-point nonlocal boundary condition is considered in this paper. These problems with nonlocal boundary condition are not self-adjoint, so the spectrum has complex points. We investigate how the spectrum in the complex plane of these problems (and for the Finite-Difference Schemes) depends on parameters γ  and ξ  of the nonlocal boundary conditions.

scholarly journals SPECTRUM CURVES FOR STURM–LIOUVILLE PROBLEM WITH INTEGRAL BOUNDARY CONDITION

2015 ◽  
Vol 20 (6) ◽  
pp. 802-818 ◽  
Author(s):  
Agnė Skučaitė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Characteristic Function ◽  
Critical Points ◽  
Nonlocal Condition ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Integral Boundary ◽  
Sturm Liouville

We consider Sturm–Liouville problem with one integral type nonlocal boundary condition depending on three parameters γ (multiplier in nonlocal condition), ξ1, ξ2 ([ξ1, ξ2] is a domain of integration). The distribution of zeroes, poles, and constant eigenvalue points of Complex Characteristic Function is presented. We investigate how Spectrum Curves depend on the parameters of nonlocal boundary conditions. In this paper we describe the behaviour of Spectrum Curves and classify critical points of Complex-Real Characteristic function. Phase Trajectories of critical points in Phase Space of the parameters ξ1, ξ2 are investigated. We present the results of modelling and computational analysis and illustrate those results with graphs.

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