Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Numerical Solution of Highly Oscillatory Differential Equations by Magnus Series Method(Izmir Institute of Technology, 2006) Kanat, Bengi; Tanoğlu, Gamze; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this study, the differential equation known as Lie-type equation where the solutions of the equation stay in the Lie-Group is considered. The solution of this equation can be represented as an infinite series whose terms consist of integrals and commutators, based on the Magnus Series. This expansion is used as a numerical geometrical integrator called Magnus Series Method, to solve this type of equations. This method which is also one of the Lie-Group methods, has slower error accumulation and more efficient computation results during the long time interval than classical numerical methods such as Runge-Kutta, since it preserves the qualitative features of the exact solutions. Several examples are considered including linear and nonlinear oscillatory problems to illustrate the efficiency of the method.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.