Investigation of a discrete Sturm–Liouville problem with two-point nonlocal boundary condition and natural approximation of a derivative in boundary condition
Abstract
The article investigates a discrete Sturm–Liouville problem with one natural boundary condition 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.
Keyword : discrete Sturm–Liouville problem, natural condition, nonlocal two-point condition, spectrum curves
How to Cite
Bingelė, K., & Štikonas, A. (2024). Investigation of a discrete Sturm–Liouville problem with two-point nonlocal boundary condition and natural approximation of a derivative in boundary condition. Mathematical Modelling and Analysis, 29(2), 309–330. https://doi.org/10.3846/mma.2024.19829
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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K. Bingelė, A. Bankauskienė and A. Štikonas. Investigation of spectrum curves for a Sturm–Liouville problem with two-point nonlocal boundary conditions. Math. Model. Anal., 25(1):53–70, 2020. https://doi.org/10.3846/mma.2020.10787
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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. https://doi.org/10.1090/qam/160437
R. Čiegis and O. Suboč. High order compact finite difference schemes on nonuniform grids. Appl. Numer. Math., 132:205–218, 2018. https://doi.org/10.1016/j.apnum.2018.06.003
R. Čiegis and N. Tumanova. On construction and analysis of finite difference schemes for pseudoparabolic problems with nonlocal boundary conditions. Math. Model. Anal., 19(2):281–297, 2014. https://doi.org/10.3846/13926292.2014.910562
W. Day. Heat Conduction within Linear Thermoelasticity. Springer-Verlag, 1985. https://doi.org/10.1007/978-1-4613-9555-3
A. Ercan. Comparative analysis for fractional nonlinear Sturm–Liouville equations with singular and non-singular kernels. AIMS Mathematics, 7(7):13325– 13343, 2022. https://doi.org/10.3934/math.2022736
C.T. Fulton. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Edinb. Math. Soc. A, 77(3–4):293– 308, 1977. https://doi.org/10.1017/S030821050002521X
C. Gao and R. Ma. Eigenvalues of discrete Sturm–Liouville problems with eigenparameter dependent boundary conditions. Linear Algebra Appl., 503:100–119, 2016. https://doi.org/10.1016/j.laa.2016.03.043
C. Gao, Y. Wang and L. Lv. Spectral properties of discrete Sturm–Liouville problems with two squared eigenparameter-dependent boundary conditions. Acta Mathematica Scientia, 40:755–781, 2020. https://doi.org/10.1007/s10473-020-0312-5
N.J. Guliyev. Essentially isospectral transformations and their applications. Ann. Mat. Pura Appl., 199(4):1621–1648, 2020. https://doi.org/10.1007/s10231-019-00934-w
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
J.C. Mason and D.C. Handscomb. Chebyshev Polynomials. Chapman&Hall/CRC, Basel, Boston, Berlin, 2003. https://doi.org/10.1201/9781420036114
J. Novickij and A. Štikonas. On the stability of a weighted finite difference scheme for wave equation with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 19(3):460–475, 2014. https://doi.org/10.15388/NA.2014.3.10
S. Pečiulytė and A. Štikonas. On positive eigenfunctions of Sturm–Liouville problem with nonlocal two-point boundary condition. Math. Model. Anal., 12(2):215– 226, 2007. https://doi.org/10.3846/1392-6292.2007.12.215-226
S. Pečiulytė, O. Štikonienė and A. Štikonas. Investigation of negative critical points of the characteristic function for problems with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 13(4):467–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
S. Roman and A. Štikonas. Third-order linear differential equation with three additional conditions and formula for Green’s function. Lith. Math. J., 50(4):426– 446, 2010. https://doi.org/10.1007/s10986-010-9097-x
S. Roman and A. Štikonas. Green’s function for discrete second-order problems with nonlocal boundary conditions. Bound. Value Probl., 2011(Article ID 767024):1–23, 2011. https://doi.org/10.1155/2011/767024
A.A. Samarskii. The Theory of Difference Schemes. Marcel Dekker, Inc., New York, Basel, 2001.
A.A. Samarskii and E.S. Nikolaev. Numerical Methods for Grid Equations, Vol. I. Direct Methods. Birkhäuser Verlag, Basel, Boston, Berlin, 1989. https://doi.org/10.1007/978-3-0348-9272-8.
M. Sapagovas, K. Pupalaigė, R. Čiupaila and T. Meškauskas. On the spectrum structure for one difference eigenvalue problem with nonlocal boundary conditions. Math. Model. Anal., 28(3):522–541, 2023. https://doi.org/10.3846/mma.2023.17503
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
M.A. Snyder. Chebyshev Methods in Numerical Approximation. Prentice–Hall, Englewood Cliffs, NJ., 1966.
A. Štikonas and S. Roman. Stationary problems with two additional conditions and formulae for Green’s functions. Numer. Funct. Anal. Optim., 30(9):1125– 1144, 2009. https://doi.org/10.1080/01630560903420932
A. Štikonas and E. Sen. Asymptotic analysis of Sturm–Liouville problem with Neumann and nonlocal two-point boundary conditions. Lith. Math. J, 62(4):519– 541, 2022. https://doi.org/10.1007/s10986-022-09577-6
A. Štikonas and E. Sen. Asymptotic analysis of Sturm–Liouville problem with Dirichlet and nonlocal two-point boundary conditions. Math. Model. Anal., 28(2):308–329, 2023. https://doi.org/10.3846/mma.2023.17617
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
E.C. Titchmarsh. Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford, 1946.
K. Bingelė, A. Bankauskienė and A. Štikonas. Investigation of spectrum curves for a Sturm–Liouville problem with two-point nonlocal boundary conditions. Math. Model. Anal., 25(1):53–70, 2020. https://doi.org/10.3846/mma.2020.10787
A. Boumenir. Eigenvalues of periodic Sturm–Liouville problems by the Shannon–Whittaker sampling theorem. Math. Comput., 68(227):1057–1066, 1999. https://doi.org/10.1090/S0025-5718-99-01053-4
J.R. Cannon. The solution of the heat equation subject to specification of energy. Quart. Appl. Math., 21(2):155–160, 1963. https://doi.org/10.1090/qam/160437
R. Čiegis and O. Suboč. High order compact finite difference schemes on nonuniform grids. Appl. Numer. Math., 132:205–218, 2018. https://doi.org/10.1016/j.apnum.2018.06.003
R. Čiegis and N. Tumanova. On construction and analysis of finite difference schemes for pseudoparabolic problems with nonlocal boundary conditions. Math. Model. Anal., 19(2):281–297, 2014. https://doi.org/10.3846/13926292.2014.910562
W. Day. Heat Conduction within Linear Thermoelasticity. Springer-Verlag, 1985. https://doi.org/10.1007/978-1-4613-9555-3
A. Ercan. Comparative analysis for fractional nonlinear Sturm–Liouville equations with singular and non-singular kernels. AIMS Mathematics, 7(7):13325– 13343, 2022. https://doi.org/10.3934/math.2022736
C.T. Fulton. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Edinb. Math. Soc. A, 77(3–4):293– 308, 1977. https://doi.org/10.1017/S030821050002521X
C. Gao and R. Ma. Eigenvalues of discrete Sturm–Liouville problems with eigenparameter dependent boundary conditions. Linear Algebra Appl., 503:100–119, 2016. https://doi.org/10.1016/j.laa.2016.03.043
C. Gao, Y. Wang and L. Lv. Spectral properties of discrete Sturm–Liouville problems with two squared eigenparameter-dependent boundary conditions. Acta Mathematica Scientia, 40:755–781, 2020. https://doi.org/10.1007/s10473-020-0312-5
N.J. Guliyev. Essentially isospectral transformations and their applications. Ann. Mat. Pura Appl., 199(4):1621–1648, 2020. https://doi.org/10.1007/s10231-019-00934-w
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
J.C. Mason and D.C. Handscomb. Chebyshev Polynomials. Chapman&Hall/CRC, Basel, Boston, Berlin, 2003. https://doi.org/10.1201/9781420036114
J. Novickij and A. Štikonas. On the stability of a weighted finite difference scheme for wave equation with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 19(3):460–475, 2014. https://doi.org/10.15388/NA.2014.3.10
S. Pečiulytė and A. Štikonas. On positive eigenfunctions of Sturm–Liouville problem with nonlocal two-point boundary condition. Math. Model. Anal., 12(2):215– 226, 2007. https://doi.org/10.3846/1392-6292.2007.12.215-226
S. Pečiulytė, O. Štikonienė and A. Štikonas. Investigation of negative critical points of the characteristic function for problems with nonlocal boundary conditions. Nonlinear Anal. Model. Control, 13(4):467–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
S. Roman and A. Štikonas. Third-order linear differential equation with three additional conditions and formula for Green’s function. Lith. Math. J., 50(4):426– 446, 2010. https://doi.org/10.1007/s10986-010-9097-x
S. Roman and A. Štikonas. Green’s function for discrete second-order problems with nonlocal boundary conditions. Bound. Value Probl., 2011(Article ID 767024):1–23, 2011. https://doi.org/10.1155/2011/767024
A.A. Samarskii. The Theory of Difference Schemes. Marcel Dekker, Inc., New York, Basel, 2001.
A.A. Samarskii and E.S. Nikolaev. Numerical Methods for Grid Equations, Vol. I. Direct Methods. Birkhäuser Verlag, Basel, Boston, Berlin, 1989. https://doi.org/10.1007/978-3-0348-9272-8.
M. Sapagovas, K. Pupalaigė, R. Čiupaila and T. Meškauskas. On the spectrum structure for one difference eigenvalue problem with nonlocal boundary conditions. Math. Model. Anal., 28(3):522–541, 2023. https://doi.org/10.3846/mma.2023.17503
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
M.A. Snyder. Chebyshev Methods in Numerical Approximation. Prentice–Hall, Englewood Cliffs, NJ., 1966.
A. Štikonas and S. Roman. Stationary problems with two additional conditions and formulae for Green’s functions. Numer. Funct. Anal. Optim., 30(9):1125– 1144, 2009. https://doi.org/10.1080/01630560903420932
A. Štikonas and E. Sen. Asymptotic analysis of Sturm–Liouville problem with Neumann and nonlocal two-point boundary conditions. Lith. Math. J, 62(4):519– 541, 2022. https://doi.org/10.1007/s10986-022-09577-6
A. Štikonas and E. Sen. Asymptotic analysis of Sturm–Liouville problem with Dirichlet and nonlocal two-point boundary conditions. Math. Model. Anal., 28(2):308–329, 2023. https://doi.org/10.3846/mma.2023.17617
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
E.C. Titchmarsh. Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford, 1946.