Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Two-Particle Schrodinger Operators With Point Interactions(01. Izmir Institute of Technology, 2020) Kızılkaya, Melih; Erman, Fatih; Erman, Fatih; 01. Izmir Institute of Technology; 04.02. Department of Mathematics; 04. Faculty of ScienceIn this thesis, a singular quantum mechanical problem, where two particles interact with each other through Dirac delta potentials in the plane, has been considered. The proof for the existence of a self-adjoint Hamiltonian operator for the model is given by using some operator theory techniques and renormalization idea in quantum field theory. Moreover, some necessary background for unbounded operators is reviewed in order to make the thesis as self-contained as possible.Master Thesis Operator Splitting Methhods for Differential Equations(Izmir Institute of Technology, 2010) Yazıcı, Yeşim; Tanoğlu, Gamze; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this thesis, consistency and stability analysis of the traditional operator splitting methods are studied. We concentrate on how to improve the classical operator splitting methods via Zassenhaus product formula. In our approach, acceleration of the initial conditions and weighted polynomial ideas for each cases are individually handled and relevant algorithms are obtained. A new higher order operator splitting methods are proposed by the means of Zassenhaus product formula and rederive the consistency bound for traditional operator splitting methods. For unbounded operators, consistency analysis are proved by the C0-semigroup approach. We adapted the Von-Neumann stability analysis to operator splitting methods. General approach to use Von-Neumann stability analysis are discussed for the operator splitting methods. The proposed operator splitting methods and traditional operator splitting methods are applied to various ODE and PDE problems.Master Thesis Operator Splitting Methods for Non-Autonomous Differential Equations(Izmir Institute of Technology, 2011) Korkut, Sıla Övgü; Tanoğlu, Gamze; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this thesis, convergency and stability analysis are studied for the non-autonomous differential equations. Not only classical operator splitting methods; Lie Trother splitting, symmetrically weighted splitting and Strang splitting but also iterative splitting method which is recent popular technique of operator splitting methods are considered. We concentrate on how to improve the operator splitting methods with the help of the Magnus expansion. In addition, we construct a new symmetric iterative splitting scheme. Then, we also study its convergence properties by using the concepts of stability, consistency and order. For this purpose, we use C0 semigroup techniques. Finally, several numerical examples are illustrated in order to confirm our theoretical results by comparing the new symmetric iterative splitting method with frequently used operator splitting methods.