Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Qualitative Properties of Solutions of Some Keller-Segel Type Systems(01. Izmir Institute of Technology, 2022) Batal, Ahmet; Özsarı, Türker; Batal, Ahmet; Özsarı, Türker; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThe main objective of this thesis is to summarize results related with solutions of some Keller - Segel type systems, which model chemotaxis. This work surveys mathematical studies starting with the work that first presented these systems in 1970. This study emphasizes the local and global existence of solutions of Keller - Segel type systems, in particular the boundedness and blow-up of solutions.Master Thesis The Dirichlet Problem for the Fractional Laplacian(Izmir Institute of Technology, 2017) Alkın, Aykut; Özsarı, Türker; Özsarı, Türker; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThis thesis is an introduction to the fractional Sobolev spaces and the fractional Laplace operator. We define the fractional Sobolev spaces and give their properties by comparing them with the classical version of Sobolev spaces. After giving the motivation that comes from the random walk theory, we define the fractional Laplacian. We focus on the mean-value property of s-harmonic functions and get into details of extension and maximum principle of the weak solution of the Dirichlet problem for the fractional Laplacian. Afterall, we explain the regularity of the weak solution of the Dirichlet problem for the fractional Laplacian inside a domain and up to the boundary, respectively.Master Thesis Boundary Controller and Observer Design for Korteweg-De Vries Type Equations(Izmir Institute of Technology, 2017) Arabacı, Eda; Özsarı, Türker; Özsarı, Türker; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThis thesis studies the back-stepping boundary controllability of Korteweg-de Vries (KdV) type equations posed on a bounded interval. The results on the back-stepping controllability of the KdV equation obtained in Cerpa and Coron (2013) and Cerpa (2012) are reviewed and extended to the KdV-Burgers (KdVB) equation. The stability of the KdVB equation is boosted to any desired exponential rate for sufficiently small initial data with a boundary feedback controller acting on the Dirichlet boundary condition. Moreover, the case that there is no full access to the system is considered. For these kinds of systems, an observer is constructed assuming an appropriate boundary measurement is available. The ideas about designing output feedback control for the KdV equation presented in Marx and Cerpa (2016), and Hasan (2016) are reviewed and extended to the KdVB model.