Nonlinear elliptic equation with nonlocal integral boundary condition depending on two parameters
Abstract
In this paper, the two-dimensional nonlinear elliptic equation with the boundary integral condition depending on two parameters is solved by finite difference method. The main aim of this paper is to investigate the conditions under those all eigenvalues of corresponding difference eigenvalue problem are positive. For this purpose, we investigate the spectrum structure of one-dimensional difference eigenvalue problem with integral condition. In particular, conditions of the existence and some properties of negative eigenvalue are analyzed in details.
Keyword : nonlinear elliptic equation, nonlocal boundary condition, difference eigenvalue problem, M-matrix, real eigenvalues of nonsymmetric matrix
How to Cite
Pupalaigė, K., Sapagovas, M., & Čiupaila, R. (2022). Nonlinear elliptic equation with nonlocal integral boundary condition depending on two parameters. Mathematical Modelling and Analysis, 27(4), 610–628. https://doi.org/10.3846/mma.2022.16209
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
A. Ashyralyev and E. Ozturk. On a difference scheme of second order of accuracy for the Bitsadze-Samarskii type nonlocal boundary-value problem. Bound. Value Probl., 2014(14):1–19, 2014. https://doi.org/10.1186/1687-2770-2014-14
C. Ashyralyyev, G. Akyuz and M. Dedeturk. Approximate solution for on inverse problem of multidimentional elliptic equation with multipoint nonlocal and Neumann boundary conditions. Electron. J. Differ. Equ., 2017(197):1–16, 2017.
G. Avalishvili, M. Avalishvili and D. Gordeziani. On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation. Appl. Math. Letters, 24:566–571, 2011. https://doi.org/10.1016/j.aml.2010.11.014
G. Berikelashvili. To a nonlocal generalization of the Dirichlet problem. J. Inequal. Appl., 2006:1–6, 2006. https://doi.org/10.1155/JIA/2006/93858
G.K. Berikelashvili and N. Khomeniki. On the convergence rate of a difference solution of the Poisson equation with fully nonlocal constraints. Nonlinear Anal. Model. Control, 19(3):367–381, 2014. https://doi.org/10.15388/na.2014.3.4
K. Bingelė, A. Bankauskienė and A. Štikonas. Spectrum curves for a discrete Sturm-Liouville problem with one integral boundary condition. Nonlin. Anal. Model. Control, 24:755–774, 2019. https://doi.org/10.15388/NA.2019.5.5
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. Economical difference schemes for the solution of a two-dimensional parabolic problem with an integral condition. Differ. Equ., 41(7):1025–1029, 2005. https://doi.org/10.1007/s10625-005-0244-9
R. Čiegis and N. Tumanova. Numerical solution of parabolic problems with nonlocal boundary conditions. Numer. Funct. Anal. Optim., 31(12):1318–1329, 2010.
R. Čiupaila, K. Pupalaigė and M. Sapagovas. On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions. Nonlin. Anal. Model. Control, 26(4):738–758, 2021. https://doi.org/10.15388/namc.2021.26.23929
R. Čiupaila, M. Sapagovas and K. Pupalaigė. M-matrices and convergence of finite difference scheme for parabolic equation with an integral boundary condition. Math. Model. Anal., 25(2):167–183, 2020. https://doi.org/10.3846/mma.2020.8023
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
A. Dosiyev. Difference method of fourth order accuracy for the Laplace equation with multilevel nonlocal conditions. J. Comput. Appl. Math., 354:587–596, 2019. https://doi.org/10.1016/j.cam.2018.04.046
A. Dosiyev and R. Reis. A fourth-order accurate difference Dirichlet problem for the approximate solution of Laplace’s equation with integral boundary condition. Advanc. Difference Equat., 2019(340):1–15, 2019. https://doi.org/10.1186/s13662-019-2282-2
A. Elsaid, S.M. Hedal and A.M.A. El-Sayed. The eigenvalue problem for elliptic differential equation with two-point nonlocal conditions. J. Appl. Anal. Comput, 5:146–158, 2015. https://doi.org/10.11948/2015013
J. Gao, D. Sun and M. Zhang. Structure of eigenvalues of multi-point boundary value problems. Advan. Difference Equat., 2010(381932):1–18, 2010. https://doi.org/10.1155/2010/381932
A.V. Gulin. Stability of nonlocal difference schemes in a subspace. Differ. Equ., 48(7):940–949, 2012. https://doi.org/10.1134/S0012266112070051
V.A. Il’in and E.I. Moiseev. Two-dimensional nonlocal boundary value problem for Poisson’s operator in differential and difference variants. Mat. Model., 2:139– 156, 1990. (In Russian)
N.I. Ionkin. Solution of one boundary value problem of the theory of heat conduction with a nonclasical boundary condition. Differ. Equ., 13:204–211, 1977.
K. Jakubėlienė, R. Čiupaila and M. Sapagovas. Semi–implicit difference scheme for a two–dimensional parabolic equation with an integral boundary condition. Math. Model. Anal., 22(5):617–633, 2017. https://doi.org/10.3846/13926292.2017.1342709
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. 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. Sturm-Liouville problem for stationary differential operator with nonlocal integral boundary condition. Math. Model. Anal., 10(4):377–392, 2005. https://doi.org/10.3846/13926292.2005.9637295
S. Pečiulytė, O. Štikonienė and A. Štikonas. Investigation of negative critical point of the characteristic function for problems with nonlocal boundary conditions. Nonlin. Anal. Model. Control, 13(4):464–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
M. Sapagovas. The eigenvalues of some problems with a nonlocal condition. Differ. Equ., 38(7):1020–1026, 2002. https://doi.org/10.1023/A:1021115915575
M. Sapagovas. On stability of finite-difference schemes for one-dimensional parabolic equations subject to integral conditions. Zh. Obchysl. Prykl. Mat., 92:70–90, 2005.
M. Sapagovas, V. Griškonienė and O. Štikonienė. Application of Mmatrices for solution of a nonlinear elliptic equation with an integral condition. Nonlinear. Anal. Model. Control, 22(4):489–504, 2017. https://doi.org/10.15388/na.2017.4.5
M. Sapagovas, A. Štikonas and O. Štikonienė. Alternating direction method for the Poisson equation with variable weight coefficients in an integral condition. Differ. Equ., 47(8):1176–1187, 2011. https://doi.org/10.1134/S0012266111080118
M. Sapagovas, R. Čiupaila, K. Jakubėlienė and S. Rutkauskas. A new eigenvalue problem for the difference operator with nonlocal conditions. Nonlin. Anal. Model. Control, 24(3):462–484, 2019. https://doi.org/10.15388/NA.2019.3.9
M. Sapagovas, O. Štikonienė, K. Jakubėlienė and R. Čiupaila. Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions. Bound. Value Probl., 2019(94):1–16, 2019. https://doi.org/10.1186/s13661-019-1202-4
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. 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. Control, 26(5):969–991, 2021. https://doi.org/10.15388/namc.2021.26.24299
O. Štikonienė, M. Sapagovas and R. Čiupaila. On iterative methods for some elliptic equations with nonlocal conditions. Nonlinear. Anal. Model. Control, 19(3):517–535, 2014. https://doi.org/10.15388/na.2014.3.14
R.S. Varga. Matrix Iterative Analysis. Prentice-Hall, 1962.
C. Ashyralyyev, G. Akyuz and M. Dedeturk. Approximate solution for on inverse problem of multidimentional elliptic equation with multipoint nonlocal and Neumann boundary conditions. Electron. J. Differ. Equ., 2017(197):1–16, 2017.
G. Avalishvili, M. Avalishvili and D. Gordeziani. On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation. Appl. Math. Letters, 24:566–571, 2011. https://doi.org/10.1016/j.aml.2010.11.014
G. Berikelashvili. To a nonlocal generalization of the Dirichlet problem. J. Inequal. Appl., 2006:1–6, 2006. https://doi.org/10.1155/JIA/2006/93858
G.K. Berikelashvili and N. Khomeniki. On the convergence rate of a difference solution of the Poisson equation with fully nonlocal constraints. Nonlinear Anal. Model. Control, 19(3):367–381, 2014. https://doi.org/10.15388/na.2014.3.4
K. Bingelė, A. Bankauskienė and A. Štikonas. Spectrum curves for a discrete Sturm-Liouville problem with one integral boundary condition. Nonlin. Anal. Model. Control, 24:755–774, 2019. https://doi.org/10.15388/NA.2019.5.5
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. Economical difference schemes for the solution of a two-dimensional parabolic problem with an integral condition. Differ. Equ., 41(7):1025–1029, 2005. https://doi.org/10.1007/s10625-005-0244-9
R. Čiegis and N. Tumanova. Numerical solution of parabolic problems with nonlocal boundary conditions. Numer. Funct. Anal. Optim., 31(12):1318–1329, 2010.
R. Čiupaila, K. Pupalaigė and M. Sapagovas. On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions. Nonlin. Anal. Model. Control, 26(4):738–758, 2021. https://doi.org/10.15388/namc.2021.26.23929
R. Čiupaila, M. Sapagovas and K. Pupalaigė. M-matrices and convergence of finite difference scheme for parabolic equation with an integral boundary condition. Math. Model. Anal., 25(2):167–183, 2020. https://doi.org/10.3846/mma.2020.8023
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
A. Dosiyev. Difference method of fourth order accuracy for the Laplace equation with multilevel nonlocal conditions. J. Comput. Appl. Math., 354:587–596, 2019. https://doi.org/10.1016/j.cam.2018.04.046
A. Dosiyev and R. Reis. A fourth-order accurate difference Dirichlet problem for the approximate solution of Laplace’s equation with integral boundary condition. Advanc. Difference Equat., 2019(340):1–15, 2019. https://doi.org/10.1186/s13662-019-2282-2
A. Elsaid, S.M. Hedal and A.M.A. El-Sayed. The eigenvalue problem for elliptic differential equation with two-point nonlocal conditions. J. Appl. Anal. Comput, 5:146–158, 2015. https://doi.org/10.11948/2015013
J. Gao, D. Sun and M. Zhang. Structure of eigenvalues of multi-point boundary value problems. Advan. Difference Equat., 2010(381932):1–18, 2010. https://doi.org/10.1155/2010/381932
A.V. Gulin. Stability of nonlocal difference schemes in a subspace. Differ. Equ., 48(7):940–949, 2012. https://doi.org/10.1134/S0012266112070051
V.A. Il’in and E.I. Moiseev. Two-dimensional nonlocal boundary value problem for Poisson’s operator in differential and difference variants. Mat. Model., 2:139– 156, 1990. (In Russian)
N.I. Ionkin. Solution of one boundary value problem of the theory of heat conduction with a nonclasical boundary condition. Differ. Equ., 13:204–211, 1977.
K. Jakubėlienė, R. Čiupaila and M. Sapagovas. Semi–implicit difference scheme for a two–dimensional parabolic equation with an integral boundary condition. Math. Model. Anal., 22(5):617–633, 2017. https://doi.org/10.3846/13926292.2017.1342709
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. 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. Sturm-Liouville problem for stationary differential operator with nonlocal integral boundary condition. Math. Model. Anal., 10(4):377–392, 2005. https://doi.org/10.3846/13926292.2005.9637295
S. Pečiulytė, O. Štikonienė and A. Štikonas. Investigation of negative critical point of the characteristic function for problems with nonlocal boundary conditions. Nonlin. Anal. Model. Control, 13(4):464–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
M. Sapagovas. The eigenvalues of some problems with a nonlocal condition. Differ. Equ., 38(7):1020–1026, 2002. https://doi.org/10.1023/A:1021115915575
M. Sapagovas. On stability of finite-difference schemes for one-dimensional parabolic equations subject to integral conditions. Zh. Obchysl. Prykl. Mat., 92:70–90, 2005.
M. Sapagovas, V. Griškonienė and O. Štikonienė. Application of Mmatrices for solution of a nonlinear elliptic equation with an integral condition. Nonlinear. Anal. Model. Control, 22(4):489–504, 2017. https://doi.org/10.15388/na.2017.4.5
M. Sapagovas, A. Štikonas and O. Štikonienė. Alternating direction method for the Poisson equation with variable weight coefficients in an integral condition. Differ. Equ., 47(8):1176–1187, 2011. https://doi.org/10.1134/S0012266111080118
M. Sapagovas, R. Čiupaila, K. Jakubėlienė and S. Rutkauskas. A new eigenvalue problem for the difference operator with nonlocal conditions. Nonlin. Anal. Model. Control, 24(3):462–484, 2019. https://doi.org/10.15388/NA.2019.3.9
M. Sapagovas, O. Štikonienė, K. Jakubėlienė and R. Čiupaila. Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions. Bound. Value Probl., 2019(94):1–16, 2019. https://doi.org/10.1186/s13661-019-1202-4
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. 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. Control, 26(5):969–991, 2021. https://doi.org/10.15388/namc.2021.26.24299
O. Štikonienė, M. Sapagovas and R. Čiupaila. On iterative methods for some elliptic equations with nonlocal conditions. Nonlinear. Anal. Model. Control, 19(3):517–535, 2014. https://doi.org/10.15388/na.2014.3.14
R.S. Varga. Matrix Iterative Analysis. Prentice-Hall, 1962.