### Study the Influence of Nonlocal Boundary Condition on the Difference Eigenvalue Problem for Elliptic Partial Differential Equation

#### Abstract

This paper presents a study of the difference eigenvalue problem for elliptic partial differential equations with a differential type multipoint nonlocal boundary conditions. We formulate the stability analysis technique which is based on the spectral structure of the transition matrix which has different types of eigenvalues. We begin by studying the one-dimensional problem and generalize the results to the two-dimensional problems by appropriate difference operators with nonlocal conditions.

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A. Ashyralyev and E. Benito, The numerical solution of the Bitsadze-Samarskii nonlocal boundary value problems with the Dirichlet-Neumann condition, Abstract and Applied Analysis 2012 (2012), ID 730804, DOI: 10.1155/2012/730804.

A. Ashyralyev and F. S. O. Tetikoglu, FDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions, Abstract and Applied Analysis 2012 (2012), ID 454831, DOI: 10.1155/2012/454831.

A. Ashyralyev and F. S. O. Tetikoglu, FDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions, Mathematical Methods in the Applied Sciences 37(17) (2014), 2663 – 2676, DOI: 10.1002/mma.3006.

A. Ashyralyev and O. Gercek, Finite difference method for multipoint nonlocal elliptic–parabolic problems, Computers and Mathematics with Applications 60(7) (2010), 2043 – 2052, DOI: 10.1016/j.camwa.2010.07.044.

A. Ashyralyev and O. Yildirim, Stable difference schemes for the hyperbolic problems subject to nonlocal boundary conditions with self-adjoint operator, Applied Mathematics and Computation 218(3) (2011), 1124 – 1131, DOI: 10.1016/j.amc.2011.03.155.

A. Ashyralyev and Y. Ozdemir, On numerical solutions for hyperbolic–parabolic equations with the multipoint nonlocal boundary condition, Journal of The Franklin Institute 351(2) (2014), 602 – 630, DOI: 10.1016/j.jfranklin.2012.08.007.

R. Ciegis, O. Šuboc and R. Ciegis, Numerical simulation of nonlocal delayed feedback controller for simple bioreactors, Informatica 29(2) (2018), 233 – 249, DOI: 10.15388/Informatica.2018.165.

R. Ciupaila, Ž. Jeseviciute and M. Sapagovas, On the eigenvalue problem for one-dimensional differential operator with nonlocal integral condition, Nonlinear Analysis: Modelling and Control 9(2) (2004), 109 – 116, DOI: 10.15388/NA.2004.9.2.15159.

M. Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Applied Numerical Mathematics 52(1) (2005), 39 – 62, DOI: 10.1016/j.apnum.2004.02.002.

A. Elsaid, S. M. Helal and A. M. A. El-Sayed, The eigenvalue problem for elliptic partial differential equation with two-point nonlocal conditions, Applied Analysis and Computation 5(1) (2015), 146 – 158, DOI: 10.11948/2015013.

A. M. A. El-Sayed, M. S. El-Azab, A. Elsaid and S. M. Helal, Eigenvalue problem for elliptic partial differential equations with nonlocal boundary conditions, Fractional Calculus and Applications 5(3S)(14) (2014), 1 – 11, URL: https://www.researchgate.net/profile/Issam_Kaddoura/post/Could_you_provide_a_reference_on_eigenvalue_problems_in_PDEs/attachment/5e43d2963843b06506da2377/AS%3A857700430786563%401581503126358/download/14_Vol.+5%283S%29.+Aug.+4%2C+2014%2C+No.14%2C+pp.+1-11..pdf.

V. A. Il’in and E. I. Moiseev, Two-dimensional nonlocal boundary value problem for Poisson’s operator in differential and difference variants, Mathematical Model 2 (1990), 132 – 156 (in Russian), URL: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mm&paperid=2433.

G. Infante, Eigenvalues and positive solutions of ODEs involving integral boundary conditions, Discrete and Continuous Dynamical Systems Supplement Volume (2005), 436 – 442, URL: https://www.aimsciences.org/article/exportPdf?id=0cb53066-c09e-43d2-a380-b3a645594b3f.

G. Infante, Eigenvalues of some non-local boundary-value problems, Proceedings of the Edinburgh Mathematical Society 46 (2003), 75 – 86, DOI: 10.1017/S0013091501001079.

F. Ivanauskas, V. Laurinavicius, M. Sapagovas and A. Neciporenko, Reaction–diffusion equation with nonlocal boundary condition subject to PID-controlled bioreactor, Nonlinear Analysis: Modelling and Control 22(2) (2017), 261 – 272, DOI: 10.15388/NA.2017.2.8.

V. L. Makarov, I. I. Lazurchak and B. I. Bandyrsky, Nonclassical asimptotic formula and approximation of arbitrary order of accuracy of the of eigenvalues in Sturm-Liouville problem with Bizadse-Samarsky conditions, Cybernetics and System Analysis 6 (2003), 862 – 879, DOI: 10.1023/B:CASA.0000020228.93401.b9.

M. Marin and A. Öchsner, The effect of a dipolar structure on the Holder stability in Green–Naghdi thermoelasticity, Continuum Mechanics and Thermodynamics 29(6) (2017), 1365 – 1374, DOI: 10.1007/s00161-017-0585-7.

J. Martín-Vaquero and J. Vigo-Aguiar, On the numerical solution of the heat conduction equations subject to nonlocal conditions, Applied Numerical Mathematics 59(10) (2009), 2507 – 2514, DOI: 10.1016/j.apnum.2009.05.007.

S. Mesloub, On a higher order two dimensional thermoelastic system combining a local and nonlocal boundar conditions, Electronic Journal of Qualitative Theory of Differential Equations 50 (2012), URL: http://real.mtak.hu/22593/1/p1176.pdf.

M. Sapagovas, Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions, Differential Equations 44(7) (2008), 1018 – 1028, DOI: 10.1134/S0012266108070148.

M. Sapagovas, On the stability of a finite-difference scheme for nonlocal parabolic boundaryvalue problems, Lithuanian Mathematical Journal 48(3) (2008), 339 – 356, URL: https://link.springer.com/content/pdf/10.1007/s10986-008-9017-5.

M. Sapagovas and Ž. Jeseviciute, On the stability of the finite-difference schemes for parabolic equations subject to integral conditions with applications for thermoelasticity, Computational Methods in Applied Mathematics 8(4) (2008), 360 – 373, DOI: 10.2478/cmam-2008-0026.

M. Sapagovas and A. Stikonas, On the structure of the spectrum of a differential operator with a nonlocal condition, Differential Equations 41(7) (2005), 1010 – 1018, URL: https://link.springer.com/content/pdf/10.1007/s10625-005-0242-y.

M. P. Sapagovas, The eigenvalues of some problem with a nonlocal condition, Differential Equations 7(38) (2002), 1020 – 1026, DOI: 10.1023/A:1021115915575.

A. Stikonas and O. Stikoniene, Characteristic functions for Sturm–Liouville problems with nonlocal boundary conditions, Mathematical Modeling and Analysis 14(2) (2009), 229 – 246, DOI: 10.3846/1392-6292.2009.14.229-246.

E. A. Volkov and A. A. Dosiyev, On the numerical solution of a multilevel nonlocal problem, Mediterranean Journal of Mathematics 13 (2016), 3589 – 3604, DOI: 10.1007/s00009-016-0704-x.

O. Yildirim and M. Uzun, On the numerical solutions of high order stable difference schemes for the hyperbolic multipoint nonlocal boundary value problems, Applied Mathematics and Computation 254 (2015), 210 – 218, DOI: 10.1016/j.amc.2014.12.117.

DOI: http://dx.doi.org/10.26713%2Fjims.v12i3.1408

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