# FINITE DIFFERENCE APPROXIMATIONS OF VARIOUS STEKLOV EIGENVALUE PROBLEMS

2022-8-26
ÖZALP, MÜCAHİT
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.

# Suggestions

 Analytical and numerical assessments of boundary variations in Steklov eigenvalue problems Bahadır, Eylem; Türk, Önder (2023-04-01) In this study, we aim to analyze the effects of several boundary variations on the spectrum of the simplified and generalized Steklov eigenvalue problems (EVPs) in which the spectral parameter resides on the boundary. We mainly focus on assessing the errors that may occur due to the finite element discretization using elements having straight edges to a curved boundary. In this respect, we analytically and numerically analyze the influence of the change in the boundary such as in uniformly expanded discs or...
 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...
 Boundary value problems for higher order linear impulsive differential equations Uğur, Ömür; Akhmet, Marat (2006-07-01) In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered. (c) 2006 Elsevier In...
Citation Formats
M. ÖZALP, “FINITE DIFFERENCE APPROXIMATIONS OF VARIOUS STEKLOV EIGENVALUE PROBLEMS,” M.S. - Master of Science, Middle East Technical University, 2022. 