Investigation of spectrum curves for a Sturm-Liouville problem with two-point nonlocal boundary conditions
Abstract
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.
Keyword : Sturm–Liouville problem, nonlocal two-point condition, eigenvalues, critical points, spectrum curves
How to Cite
Bingelė, K., Bankauskienė, A., & Štikonas, A. (2020). Investigation of spectrum curves for a Sturm-Liouville problem with two-point nonlocal boundary conditions. Mathematical Modelling and Analysis, 25(1), 53-70. https://doi.org/10.3846/mma.2020.10787
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J.R. Cannon. The solution of the heat equation subject to specification of energy. Quart. Appl. Math., 21(2):155–160, 1963.
R. Čiegis, A. Štikonas, O. Štikonienė and O. Suboč. A monotonic finite-difference scheme for a parabolic problem with nonlocal conditions. Differ. Equ., 38(7):1027–1037, 2002. https://doi.org/10.1023/A:1021167932414
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A.O. Gelfond. Resheniye uravneniy v tselykh chislakh. Populyarnyye lektsii po matematike. Vypusk 8. Izdaniye 3-ye. Nauka, 1978. (in Russian)
A.V. Gulin, N.I. Ionkin and V.A. Morozova. Stability of a nonlocal two-dimensional finite-difference problem. Differ. Equ., 37(7):960–978, 2001.
A.V. Gulin and V.A. Morozova. On the stability of a nonlocal finite-difference boundary value problem. Differ. Equ., 39(7):962–967, 2003. https://doi.org/10.1023/B:DIEQ.0000009192.30909.13
V.A. Il’in. On a connection between the form of the boundary conditions and the basis property and the property of equiconvergence with a trigonometric series of expansions in root functions of a nonself-adjoint differential operator. Differ. Equ., 30(9):1516–1529, 1994.
N.I. Ionkin. A problem for the heat equation with a nonclassical (nonlocal) boundary condition. Technical Report 14, Numerikus Modzerek, Budapest, 1979.
T. Leonavičienė, A. Bugajev, G. Jankevičiūtė and R. Čiegis. On stability analysis of finite difference schemes for generalized Kuramoto–Tsuzuki equation with nonlocal boundary conditions. Math. Model. Anal., 21(5):630–643, 2016. https://doi.org/10.3846/13926292.2016.1198836
S. Pečiulytė and A. Štikonas. Sturm–Liouville problem for stationary differential operator with nonlocal two-point boundary conditions. Nonlinear Anal. Model. Control, 11(1):47–78, 2006. https://doi.org/10.15388/NA.2006.11.1.14764
S. Pečiulytė and A. Štikonas. Distribution of the critical and other points in boundary problems with nonlocal boundary condition. Liet. Mat. Rink., 47(Special Issue):405–410, 2007. https://doi.org/10.15388/LMR.2007.sm01
S. Pečiulytė, O. Štikonienė and A. Štikonas. Investigation of negative critical point of the characteristic function for problems with nonlocal boundary condition. Nonlinear Anal. Model. Control, 13(4):467–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
M.P. Sapagovas. A boundary value problem with a nonlocal condition for a system of ordinary differential equations. Differ. Equ., 36(7):1078–1085, 2000. https://doi.org/10.1007/BF02754510
M.P. Sapagovas. The eigenvalues of some problem with a nonlocal condition. Differ. Equ., 38(7):1020–1026, 2002. https://doi.org/10.1023/A:1021115915575
M.P. Sapagovas and A.D. Štikonas. On the structure of the spectrum of a differential operator with a nonlocal condition. Differ. Equ., 41(7):1010–1018, 2005. https://doi.org/10.1007/s10625-005-0242-y
A.A. Shkalikov. Bases formed by eigenfunctions of ordinary differential operators with integral boundary conditions. Vestnik Moskovsk. Universit., Ser. 1 Matem. Mechan, 6:12–21, 1982. (in Russian)
A. Skučaitė. Investigation of the spectrum for Sturm–Liouville problem with a nonlocal integral boundary condition. PhD thesis, Vilnius University, 2016.
A. Skučaitė, K. Skučaitė-Bingelė, S. Pečiulytė and A. Štikonas. Investigation of the spectrum for the Sturm–Liouville problem with one integral boundary condition. Nonlinear Anal. Model. Control, 15(4):501–512, 2010. https://doi.org/10.15388/NA.15.4.14321
A. Skučaitė and A. Štikonas. Spectrum curves for Sturm–Liouville problem with integral boundary condition. Math. Model. Anal., 20(6):802–818, 2015. https://doi.org/10.3846/13926292.2015.1116470
A. Štikonas. The Sturm–Liouville problem with a nonlocal boundary condition. Lith. Math. J.,47(3):336–351,2007. https://doi.org/10.1007/s10986-007-0023-9
A. Štikonas. Investigation of characteristic curve for Sturm–Liouville problem with nonlocal boundary conditions on torus. Math. Model. Anal., 16(1):1–22, 2011. https://doi.org/10.3846/13926292.2011.552260
A. Štikonas. A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 19(3):301–334, 2014. https://doi.org/10.15388/NA.2014.3.1
A. Štikonas and O. Štikonienė. Characteristic functions for Sturm–Liouville problems with nonlocal boundary conditions. Math. Model. Anal., 14(2):229–246, 2009. https://doi.org/10.3846/1392-6292.2009.14.229-246
A.V. Bitsadze and A.A. Samarskii. Some elementary generalizations of linear elliptic boundary value problems. Soviet Math. Dokl., 10:398–400, 1969.
J.R. Cannon. The solution of the heat equation subject to specification of energy. Quart. Appl. Math., 21(2):155–160, 1963.
R. Čiegis, A. Štikonas, O. Štikonienė and O. Suboč. A monotonic finite-difference scheme for a parabolic problem with nonlocal conditions. Differ. Equ., 38(7):1027–1037, 2002. https://doi.org/10.1023/A:1021167932414
R. Čiupaila, Ž. Jesevičiūtė and M. Sapagovas. On the eigenvalue problem for one-dimensional differential operator with nonlocal integral condition. Nonlinear Anal. Model. Control, 9(2):109–116, 2004. Available from Internet: http://www.mii.lt/na/issues/NA_0902/NA09201.pdf
A.O. Gelfond. Resheniye uravneniy v tselykh chislakh. Populyarnyye lektsii po matematike. Vypusk 8. Izdaniye 3-ye. Nauka, 1978. (in Russian)
A.V. Gulin, N.I. Ionkin and V.A. Morozova. Stability of a nonlocal two-dimensional finite-difference problem. Differ. Equ., 37(7):960–978, 2001.
A.V. Gulin and V.A. Morozova. On the stability of a nonlocal finite-difference boundary value problem. Differ. Equ., 39(7):962–967, 2003. https://doi.org/10.1023/B:DIEQ.0000009192.30909.13
V.A. Il’in. On a connection between the form of the boundary conditions and the basis property and the property of equiconvergence with a trigonometric series of expansions in root functions of a nonself-adjoint differential operator. Differ. Equ., 30(9):1516–1529, 1994.
N.I. Ionkin. A problem for the heat equation with a nonclassical (nonlocal) boundary condition. Technical Report 14, Numerikus Modzerek, Budapest, 1979.
T. Leonavičienė, A. Bugajev, G. Jankevičiūtė and R. Čiegis. On stability analysis of finite difference schemes for generalized Kuramoto–Tsuzuki equation with nonlocal boundary conditions. Math. Model. Anal., 21(5):630–643, 2016. https://doi.org/10.3846/13926292.2016.1198836
S. Pečiulytė and A. Štikonas. Sturm–Liouville problem for stationary differential operator with nonlocal two-point boundary conditions. Nonlinear Anal. Model. Control, 11(1):47–78, 2006. https://doi.org/10.15388/NA.2006.11.1.14764
S. Pečiulytė and A. Štikonas. Distribution of the critical and other points in boundary problems with nonlocal boundary condition. Liet. Mat. Rink., 47(Special Issue):405–410, 2007. https://doi.org/10.15388/LMR.2007.sm01
S. Pečiulytė, O. Štikonienė and A. Štikonas. Investigation of negative critical point of the characteristic function for problems with nonlocal boundary condition. Nonlinear Anal. Model. Control, 13(4):467–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
M.P. Sapagovas. A boundary value problem with a nonlocal condition for a system of ordinary differential equations. Differ. Equ., 36(7):1078–1085, 2000. https://doi.org/10.1007/BF02754510
M.P. Sapagovas. The eigenvalues of some problem with a nonlocal condition. Differ. Equ., 38(7):1020–1026, 2002. https://doi.org/10.1023/A:1021115915575
M.P. Sapagovas and A.D. Štikonas. On the structure of the spectrum of a differential operator with a nonlocal condition. Differ. Equ., 41(7):1010–1018, 2005. https://doi.org/10.1007/s10625-005-0242-y
A.A. Shkalikov. Bases formed by eigenfunctions of ordinary differential operators with integral boundary conditions. Vestnik Moskovsk. Universit., Ser. 1 Matem. Mechan, 6:12–21, 1982. (in Russian)
A. Skučaitė. Investigation of the spectrum for Sturm–Liouville problem with a nonlocal integral boundary condition. PhD thesis, Vilnius University, 2016.
A. Skučaitė, K. Skučaitė-Bingelė, S. Pečiulytė and A. Štikonas. Investigation of the spectrum for the Sturm–Liouville problem with one integral boundary condition. Nonlinear Anal. Model. Control, 15(4):501–512, 2010. https://doi.org/10.15388/NA.15.4.14321
A. Skučaitė and A. Štikonas. Spectrum curves for Sturm–Liouville problem with integral boundary condition. Math. Model. Anal., 20(6):802–818, 2015. https://doi.org/10.3846/13926292.2015.1116470
A. Štikonas. The Sturm–Liouville problem with a nonlocal boundary condition. Lith. Math. J.,47(3):336–351,2007. https://doi.org/10.1007/s10986-007-0023-9
A. Štikonas. Investigation of characteristic curve for Sturm–Liouville problem with nonlocal boundary conditions on torus. Math. Model. Anal., 16(1):1–22, 2011. https://doi.org/10.3846/13926292.2011.552260
A. Štikonas. A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 19(3):301–334, 2014. https://doi.org/10.15388/NA.2014.3.1
A. Štikonas and O. Štikonienė. Characteristic functions for Sturm–Liouville problems with nonlocal boundary conditions. Math. Model. Anal., 14(2):229–246, 2009. https://doi.org/10.3846/1392-6292.2009.14.229-246