The spectrum of a non-symmetric matrix A is studied, which resulting from difference approximation of the differential operators of the second order with nonlocal boundary conditions and variable coefficients. Such matrices can be both simple and double real eigenvalue and a complex conjugate pairs of eigenvalues. The original matrix A is mapped to the so-called associated matrix L – symmetric tridiagonal matrix, little differing from A . It is known that all eigenvalues of the matrix L are real and different. They can be found by the bisection method using a sequence of the Sturm’s polynomials. In the paper on the basis of numerous examples are compared the properties of the spectra of matrices L and A . The method offers of approximate calculating of eigenvalues of the matrix A .
Similar content being viewed by others
References
N. I. Ionkin, “Finite-difference schemes for a nonclassical problem,” Vest. MGU, Ser. 15: Vychisl. Matem. Kibern., No. 2, 20–32 (1977).
N. I. Ionkin, A Problem for the Heat Equation with a Nonclassical (Nonlocal) Boundary Condition, Preprint No. 14, Numerikus Modzerek, Budapest.
A. V. Gulin, N. I. Ionkin, and V. A. Morozova, Stability of Nonlocal Difference Schemes [in Russian], Izd. LKI, Moscow (2008).
A. V. Gulin, N. I. Ionkin, and V. A. Morozova, Difference Schemes for Nonstationary Nonlocal Problems [in Russian], MAKS Press, Izd. MGU, Moscow (2010).
N. I. Ionkin and E. A. Valikova, “The maximum principle for a nonlocal non-self-adjoint boundary-value problem,” Diff. Uravn., 31, No. 7, 1232–1239 (1995).
N. I. Ionkin and E. A. Valikova, “Eigenvalues and eigenfunctions of a nonclassical boundary-value problem,” Matem. Modelirovanie, 8, No. 1, 53-63 (1996).
N. I. Ionkin and N. Zidov, “C-stability of difference schemes for a nonclassical problem,” Vestn. MGU, Ser. 15: Vychisl. Matem. Kibern., No. 1, 8–16 (1982).
N. I. Ionkin, V. L. Makarov, and D. G. Furletov, “Stability and convergence in C-norm of difference schemes for a parabolic equation with nonlocal boundary conditions,” Matem. Modelirovanie, 4, No. 4, 63–73 (1992).
A. A. Alikhanov, “Nonlocal boundary-value problems in differential and finite-difference settings,” Diff. Uravn., 44, No. 7, 924–931 (2008).
A. A. Alikhanov, “Stability and convergence of nonlocal difference schemes,” Diff. Uravn., 46, No. 7, 942–954 (2010).
A. A. Samarskii, Theory of Finite-Difference Schemes [in Russian], Nauka, Moscow (1989).
S. K. Godunov, Solution of Linear Equation Systems [in Russian], Nauka, Novosibirsk (1980).
V. P. Il’in and Yu. I. Kuznetsov, Tridiagonal Matrices and Their Applications [in Russian], Nauka, Fizmatgiz (1985).
J. H. Wilkinson, The Algebraic Eigenvalue Problem [Russian translation], Nauka, Moscow (1970).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Matematika i Informatika, No. 47, 2014, pp. 25–42.
Rights and permissions
About this article
Cite this article
Gulin, A.V., Morozova, V.A. The Spectrum of a Nonlocal Difference Operator with Variable Coefficients. Comput Math Model 26, 484–500 (2015). https://doi.org/10.1007/s10598-015-9286-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10598-015-9286-x