Abstract
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.
Highlights
While applying mathematical modelling to various phenomena of physics [8,10], biology and ecology [14] there often arise problems with non-classical boundary conditions, which relate the values of unknown function on the boundary and inside of the given domain
In this paper the spectrum for Sturm–Liouville problem with one integral nonlocal boundary conditions (NBC) depending on two parameters was investigated
Qualitative view of the Spectrum Curves with respect to parameters ξ1 and ξ2 in integral BC, the location of the zeroes, poles and CE points of the Characteristic Function (CF) is very important for investigation
Summary
While applying mathematical modelling to various phenomena of physics [8,10], biology and ecology [14] there often arise problems with non-classical boundary conditions, which relate the values of unknown function on the boundary and inside of the given domain Boundary conditions of such type are called nonlocal boundary conditions (NBC). We investigate critical points of Characteristic Function (CF) and calculate the behavior real and complex parts of a spectrum. In this paper we investigate special case (q = 0) of this problem with one integral NBC, and functionals defined by the formulas y(0) = 0, ξ2 y(1) = γ y(t) dt, ξ1 with parameters γ ∈ R and ξ ∈ Sξ := {(ξ1, ξ2) ∈ [0, 1]2, ξ1 ξ2}. Our main goal is to investigate the influence of parameters γ, ξ1, ξ2 for the spectrum of Sturm–Liouville problem and a behavior of the critical points of Complex-Real Characteristic Function (CF). Negative critical points for problems with two-point or integral NBC’s with one parameter ξ were investigated in paper [21], too
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