Show/Hide Menu
Hide/Show Apps
Logout
Türkçe
Türkçe
Search
Search
Login
Login
OpenMETU
OpenMETU
About
About
Open Science Policy
Open Science Policy
Communities & Collections
Communities & Collections
Help
Help
Frequently Asked Questions
Frequently Asked Questions
Guides
Guides
Thesis submission
Thesis submission
MS without thesis term project submission
MS without thesis term project submission
Publication submission with DOI
Publication submission with DOI
Publication submission
Publication submission
Supporting Information
Supporting Information
General Information
General Information
Copyright, Embargo and License
Copyright, Embargo and License
Contact us
Contact us
FINITE DIFFERENCE APPROXIMATIONS OF VARIOUS STEKLOV EIGENVALUE PROBLEMS
Download
thesis.pdf
Date
2022-8-26
Author
ÖZALP, MÜCAHİT
Metadata
Show full item record
This work is licensed under a
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License
.
Item Usage Stats
55
views
25
downloads
Cite This
In this thesis, the finite difference method (FDM) is employed to numerically solve differently defined Steklov eigenvalue problems (EVPs) that are characterized by the existence of a spectral parameter on the whole or a part of the domain boundary. The FDM approximation of the Laplace EVP is also considered due to the fact that the defining differential operator in a Steklov EVP is the Laplace operator. The fundamentals of FDM are covered and their applications on some BVPs involving Laplace operator are discussed. Using Taylor's series expansions, approximation formulas for the derivatives of the functions are provided, with varying degrees of accuracy. To validate our formulation of FDM, the second and fourth order formulas are first used to approximate two test problems for which the analytical solutions are known, namely, the Poisson problem and the Laplace EVP. It is demonstrated that the solutions from the FDM agree well with the exact ones and that the results from the fourth order scheme are superior to those from the second order one. Secondly, we consider two Steklov eigenvalue problems that are distinct from each other by the associated boundary conditions. Specifically, the standard Steklov EVP with a mixed type boundary condition involving a spectral parameter is analyzed as the first problem, whereas in the second problem, the boundary of the computational domain is divided into two parts; one with Neumann type boundary condition and the other with spectral boundary condition. The discretization of the problem is performed by several orders of finite difference formulas for the first time to the best of our knowledge. The agreement between the approximate and exact eigenfunctions is shown using contour plots, and the rate of convergence of the approximate eigenvalues to the reference ones is given. It has been noted that the use of higher order finite difference approximations -of at least second order- for not only the differential equation but also the boundary conditions advances the rate of convergence. Consequently, the present study demonstrates how a second-order convergence can be acquired by the application of fourth-order finite difference formulas for both the differential operator and the accompanying boundary conditions.
Subject Keywords
Steklov eigenvalue problems
,
Laplace eigenvalue problem
,
finite difference method
,
second and fourth order schemes
,
Steklov özdeger problemleri
,
Laplace özdeğer problemi
,
sonlu farklar metodu
,
ikinci ve dördüncü mertebeden şemalar
URI
https://hdl.handle.net/11511/98759
Collections
Graduate School of Natural and Applied Sciences, Thesis
Suggestions
OpenMETU
Core
Finite bisimulations for switched linear systems
Aydın Göl, Ebru; Lazar, Mircea; Belta, Calin (2013-02-04)
In this paper, we consider the problem of constructing a finite bisimulation quotient for a discrete-time switched linear system in a bounded subset of its state space. Given a set of observations over polytopic subsets of the state space and a switched linear system with stable subsystems, the proposed algorithm generates the bisimulation quotient in a finite number of steps with the aid of sublevel sets of a polyhedral Lyapunov function. Starting from a sublevel set that includes the origin in its interio...
Finite Bisimulations for Switched Linear Systems
Aydın Göl, Ebru; Lazar, Mircea; Belta, Calin (2014-12-01)
In this paper, we consider the problem of constructing a finite bisimulation quotient for a discrete-time switched linear system in a bounded subset of its state space. Given a set of observations over polytopic subsets of the state space and a switched linear system with stable subsystems, the proposed algorithm generates the bisimulation quotient in a finite number of steps with the aid of sublevel sets of a polyhedral Lyapunov function. Starting from a sublevel set that includes the origin in its interio...
Near-field performance analysis of locally-conformal perfectly matched absorbers via Monte Carlo simulations
Ozgun, Ozlem; Kuzuoğlu, Mustafa (2007-12-10)
In the numerical solution of some boundary value problems by the finite element method (FEM), the unbounded domain must be truncated by an artificial absorbing boundary or layer to have a bounded computational domain. The perfectly matched layer (PML) approach is based on the truncation of the computational domain by a reflectionless artificial layer which absorbs outgoing waves regardless of their frequency and angle of incidence. In this paper, we present the near-field numerical performance analysis of o...
EXACT SPIN AND PSEUDO-SPIN SYMMETRIC SOLUTIONS OF THE DIRAC-KRATZER PROBLEM WITH A TENSOR POTENTIAL VIA LAPLACE TRANSFORM APPROACH
Arda, Altug; Sever, Ramazan (2012-09-28)
Exact bound state solutions of the Dirac equation for the Kratzer potential in the presence of a tensor potential are studied by using the Laplace transform approach for the cases of spin- and pseudo-spin symmetry. The energy spectrum is obtained in the closed form for the relativistic as well as non-relativistic cases including the Coulomb potential. It is seen that our analytical results are in agreement with the ones given in the literature. The numerical results are also given in a table for different p...
Geometric measures of entanglement
UYANIK, KIVANÇ; Turgut, Sadi (American Physical Society (APS), 2010-03-01)
The geometric measure of entanglement, which expresses the minimum distance to product states, has been generalized to distances to sets that remain invariant under the stochastic reducibility relation. For each such set, an associated entanglement monotone can be defined. The explicit analytical forms of these measures are obtained for bipartite entangled states. Moreover, the three-qubit case is discussed and it is argued that the distance to the W states is a new monotone.
Citation Formats
IEEE
ACM
APA
CHICAGO
MLA
BibTeX
M. ÖZALP, “FINITE DIFFERENCE APPROXIMATIONS OF VARIOUS STEKLOV EIGENVALUE PROBLEMS,” M.S. - Master of Science, Middle East Technical University, 2022.