Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Discrete Fractional Integral Operators and Their Relations To Number Theory(Izmir Institute of Technology, 2018) Sert, Ezgi; Temur, Faruk; Temur, Faruk; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThe aim of this thesis is to get estimates on discrete fractional integral operators by using number theory. These operators, starting with the studies of Arkipov and Oskolkov, have been investigated for a long time. Fourier analysis and topics related to it have been used in these studies. However, this study will put forward new results on these operators with the help of arithmetic.Master Thesis Finite Element Based Stabilized Methods for Time Dependent Convection-Diffusion Equation and Their Analysis(Izmir Institute of Technology, 2016) Yılmaz, Kemal Cem; Tanoğlu, Gamze; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThis study is focused on a Fourier stability and accuracy analysis of the time integration algorithms using generalized trapezioidal family of methods of scalar unsteady convection–diffusion equation with periodic boundary conditions. The discretization in space dimension is performed by standard Galerkin finite element formulation for low Peclet numbers and stabilized finite element formulation for large Peclet numbers. The stability analysis is performed namely by von-Neumann stability analysis. Accuracy is measured in terms of damping errors and phase speed errors. The behaviour of these temporal errors of the particular time stepping algorithms, i.e. forward Euler, Crank-Nicolson and backward Euler methods are compared with each other. Particular attention is given to the stabilized finite element formulation, that is the case where we consider high Peclet numbers. For this case, it is concluded that the Crank-Nicolson time stepping represents a better approximate solution compared to the other time integrators on transport process of an initial wave profile. Finally, at the end of the study, we derive a stabilization parameter under a particular condition on Courant number, which provides the relative phase speed error being almost equivalent to its optimal level, that is, the waves with different Fourier modes propagate almost in the same speed. Theoretical results are confirmed by a number of numerical experiments.Master Thesis Fourier Analysis on the Lorentz Group and Relativistic Quantum Mechanics(Izmir Institute of Technology, 2008) Ok, Zahide; Kasım, Rıfat Mir; Kasım, Rıfat Mir; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThe non-relativistic Schrödinger and Lippman-Schwinger equations are described. The expressions of these equations are investigated in momentum and configuration spaces, using Fourier transformation. The plane wave, which is generating function for the matrix elements of three dimensional Euclidean group in spherical basis, expanded in terms of Legendre polynomials and spherical Bessel functions. Also explicit calculation of Green.s function is done.The matrix elements of the unitary irreducible representations of Lorentz group are used to introduce Fourier expansion of plane waves. And the kernel of Gelfand-Graev transformation, which is the relativistic plane wave, is expanded in to these matrix elements. Then relativistic differential difference equation in configuration space is constructed.Lippman-Schwinger equations are studied in Lobachevsky space (hyperbolic space). An analogous to the non-relativistic case, using the finite difference Schrödinger equation, one dimensional Green.s function is analyzed for the relativistic case . Also the finite difference analogue of the Heavyside step function is investigated.