On the spectrum structure for one difference eigenvalue problem with nonlocal boundary conditions
Abstract
The difference eigenvalue problem approximating the one-dimensional differential equation with the variable weight coefficients in an integral conditions is considered. The cases without negative eigenvalue in the spectrum of difference eigenvalue problem were analyzed. Analysis of the conditions of stability of difference schemes for parabolic equations was carried out according to the theoretical results and results of the numerical experiment.
Keyword : difference eigenvalue problem, nonlocal boundary conditions, stability of difference schemes
How to Cite
Sapagovas, M., Pupalaigė, K., Čiupaila, R., & Meškauskas, T. (2023). On the spectrum structure for one difference eigenvalue problem with nonlocal boundary conditions. Mathematical Modelling and Analysis, 28(3), 522–541. https://doi.org/10.3846/mma.2023.17503
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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G. Ekolin. Finite–difference methods for a nonlocal boundary–value problem for heat equation. BIT, 31:245–261, 1991. https://doi.org/10.1007/BF01931285
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G. Fairweather and J.C. Lopez-Marcos. Galerkin methods for a semilinear parabolic problem with nonlocal conditions. Adv. Comp. Math., 6:243–262, 1996. https://doi.org/10.1007/BF02127706
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N.I. Ionkin. Solution of one boundary value problem of the theory of heat condaction with a nonclasical boundary condition. Differ. Equ., 13:204–211, 1977.
J. Jachimavičienė, Ž. Jesevičiūtė and M. Sapagovas. The stability of finite–difference schemes for a pseudoparabolic equations with nonlocal conditions. Numer. Funct. Anal. Optim., 30(9):988–1001, 2009. https://doi.org/10.1080/01630560903405412
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J. Martin-Vaquero and J. Vigo-Aguiar. On the numerical solution of the heat conduction equations subject to nonlocal conditions. Appl. Numer. Math., 59:2507– 2514, 2009. https://doi.org/10.1016/j.apnum.2009.05.007
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, O. Štikonienė and A. Štikonas. Investigation of negative critical points of the characteristic function for problem with nonlocal boundary conditions. Nonlin. Anal. Model. Contr., 13(4):467–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
K. Pupalaigė, M. Sapagovas and R. Čiupaila. Nonlinear elliptic equation with nonlocal integral boundary condition depending on two parameters. Math. Model. Anal., 27(4):610–628, 2022. https://doi.org/10.3846/mma.2022.16209
B.P. Rynne. Spectral properties and nodal solutions for second-order boundary value problems. Nonlin. Anal. Theory, Meth. & Applic, 67(12):3318–3327, 2007. https://doi.org/10.1016/j.na.2006.10.014
M. Sapagovas. The eigenvalues of some problems with a nonlocal condition. Differ. Equ., 38:1020–1026, 2002. https://doi.org/10.1023/A:1021115915575
M. Sapagovas. On the stability of a finite-difference scheme for nonlocal parabolic boundary–value problems. Lith. Math. J., 48(3):339–356, 2008. https://doi.org/10.1007/s10986-008-9017-5
M. Sapagovas, T. Meškauskas and F. Ivanauskas. Numerical spectral analysis of a difference operator with non-local boundary conditions. Appl. Math. Comput., 218(14):7515–7527, 2012. https://doi.org/10.1016/j.amc.2012.01.017
M. Sapagovas, T. Meškauskas and F. Ivanauskas. Influence of complex coefficients on the stability of difference scheme for parabolic equations with nonlocal conditions. Appl. Math. Comp., 332:228–240, 2018. https://doi.org/10.1016/j.amc.2018.03.072
M. Sapagovas, R. Čiupaila, Ž. Jokšienė and T. Meškauskas. Computational experiment for stability analysis of difference schemes with nonlocal conditions. Informatica, 24(2):275–290, 2013. https://doi.org/10.15388/Informatica.2013.396
M.P. Sapagovas and A.D. Štikonas. On the stucture 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. 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 E. Şen. Asymptotic analysis of Sturm-Liouville problem with nonlocal integral type boundary condition. Nonlin. Anal. Model. Contr., 26(15):969–991, 2021. https://doi.org/10.15388/namc.2021.26.24299
R.S. Varga. Matrix Iterative Analysis. Prentice Hall, New Jersy, 1962.
Z.C. Zhou and F.F. Liao. Structure and asymptotic expansion of eigenvalues of an integral type nonlocal problem. Electr. J. Different. Equat, 2016(283):1–12, 2016.
K. Bingelė, A. Bankauskienė and A. Štikonas. Spectrum curves for a discrete Sturm-Lioville problem with one integral boundary condition. Nonlin. Anal. Model. Control, 24:755–774, 2019. https://doi.org/10.15388/NA.2019.5.5
B. Cahlon, D.M. Kulkarni and P. Shi. Stepwise stability for the heat equation with a nonlocal constrain. SIAM J. Numer. Anal., 32(2):571–593, 1995. https://doi.org/10.1137/0732025
R. Čiegis, A. Štikonas, O. Štikonienė and O. Suboč. A monotonic finite– difference scheme for a parabolic problem with nonlocal condition. Differ. Equ., 38(7):1027–1037, 2002. https://doi.org/10.1023/A:1021167932414
R. Čiupaila, K. Pupalaigė and M. Sapagovas. On the numerical solution for nonlinear elliptic equation with variable weight coefficients in an integral condition. Nonlinear. Anal. Model. Contr., 26(4):738–758, 2021. https://doi.org/10.15388/namc.2021.26.23929
M.R. Cui. Convergence analysis of compact difference schemes for diffusion equation with nonlocal boundary conditions. Appl. Math. Comput., 260(2015):227– 241, 2015. https://doi.org/10.1016/j.amc.2015.03.039
W.A. Day. Extensions of a property of solutions of the heat equation subject to linear thermoelasticity and other theories. Quart. Appl. Math., 40:319–330, 1982. https://doi.org/10.1090/qam/678203
M. Dehghan. Efficient techniques for the second-order parabolic equation subject to nonlocal specifications. Appl. Numer. Math., 52:39–62, 2005. https://doi.org/10.1016/j.apnum.2004.02.002
G. Ekolin. Finite–difference methods for a nonlocal boundary–value problem for heat equation. BIT, 31:245–261, 1991. https://doi.org/10.1007/BF01931285
N. El-Mowafy, S.M. Hedal and M.S. El-Hzab. Study the influence of nonlocal boundary condition on the difference eigenvalue problem for differential equation. J. Informat. and Mathem. Scienc., 12(3):209–222, 2020.
G. Fairweather and J.C. Lopez-Marcos. Galerkin methods for a semilinear parabolic problem with nonlocal conditions. Adv. Comp. Math., 6:243–262, 1996. https://doi.org/10.1007/BF02127706
J. Gao, D. Sun and M. Zhang. Structure of eigenvalues of multi-point boundary value problems. Advan. Difference Equat., 381932(2010):1–18, 2010.
N.I. Ionkin. Solution of one boundary value problem of the theory of heat condaction with a nonclasical boundary condition. Differ. Equ., 13:204–211, 1977.
J. Jachimavičienė, Ž. Jesevičiūtė and M. Sapagovas. The stability of finite–difference schemes for a pseudoparabolic equations with nonlocal conditions. Numer. Funct. Anal. Optim., 30(9):988–1001, 2009. https://doi.org/10.1080/01630560903405412
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
Y. Liu. Numerical solution of the heat equation with nonlocal boundary conditions. J. Comput. Appl. Math, 110:115–127, 1999. https://doi.org/10.1016/S0377-0427(99)00200-9
R. Ma and D. O’Regan. Nodal solution for second-order m-point boundary value problems with nonlinearities across several eigenvalues. Nonlin. Anal. Theory, Math & Appl., 64(2006):1562–1577, 2006. https://doi.org/10.1016/j.na.2005.07.007
J. Martin-Vaquero and J. Vigo-Aguiar. On the numerical solution of the heat conduction equations subject to nonlocal conditions. Appl. Numer. Math., 59:2507– 2514, 2009. https://doi.org/10.1016/j.apnum.2009.05.007
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, O. Štikonienė and A. Štikonas. Investigation of negative critical points of the characteristic function for problem with nonlocal boundary conditions. Nonlin. Anal. Model. Contr., 13(4):467–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
K. Pupalaigė, M. Sapagovas and R. Čiupaila. Nonlinear elliptic equation with nonlocal integral boundary condition depending on two parameters. Math. Model. Anal., 27(4):610–628, 2022. https://doi.org/10.3846/mma.2022.16209
B.P. Rynne. Spectral properties and nodal solutions for second-order boundary value problems. Nonlin. Anal. Theory, Meth. & Applic, 67(12):3318–3327, 2007. https://doi.org/10.1016/j.na.2006.10.014
M. Sapagovas. The eigenvalues of some problems with a nonlocal condition. Differ. Equ., 38:1020–1026, 2002. https://doi.org/10.1023/A:1021115915575
M. Sapagovas. On the stability of a finite-difference scheme for nonlocal parabolic boundary–value problems. Lith. Math. J., 48(3):339–356, 2008. https://doi.org/10.1007/s10986-008-9017-5
M. Sapagovas, T. Meškauskas and F. Ivanauskas. Numerical spectral analysis of a difference operator with non-local boundary conditions. Appl. Math. Comput., 218(14):7515–7527, 2012. https://doi.org/10.1016/j.amc.2012.01.017
M. Sapagovas, T. Meškauskas and F. Ivanauskas. Influence of complex coefficients on the stability of difference scheme for parabolic equations with nonlocal conditions. Appl. Math. Comp., 332:228–240, 2018. https://doi.org/10.1016/j.amc.2018.03.072
M. Sapagovas, R. Čiupaila, Ž. Jokšienė and T. Meškauskas. Computational experiment for stability analysis of difference schemes with nonlocal conditions. Informatica, 24(2):275–290, 2013. https://doi.org/10.15388/Informatica.2013.396
M.P. Sapagovas and A.D. Štikonas. On the stucture 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. 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 E. Şen. Asymptotic analysis of Sturm-Liouville problem with nonlocal integral type boundary condition. Nonlin. Anal. Model. Contr., 26(15):969–991, 2021. https://doi.org/10.15388/namc.2021.26.24299
R.S. Varga. Matrix Iterative Analysis. Prentice Hall, New Jersy, 1962.
Z.C. Zhou and F.F. Liao. Structure and asymptotic expansion of eigenvalues of an integral type nonlocal problem. Electr. J. Different. Equat, 2016(283):1–12, 2016.