Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Solutions of Initial and Boundary Value Problems for Inhomogeneous Burgers Equations With Time-Variable Coefficients(Izmir Institute of Technology, 2016) Bozacı, Aylin; Atılgan Büyükaşık, Şirin; Atılgan Büyükaşık, Şirin; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this thesis, we have investigated initial-boundary value problems on semiinfinite line for inhomogeneous Burgers equation with time-variable coecients. We have formulated the solutions for the cases with Dirichlet and Neumann boundary conditions. We showed that the Dirichlet problem for the variable parametric Burgers equation is solvable in terms of a linear ordinary dierential equation and a linear second kind singular Volterra integral equation. Then, for particular models with special initial and Dirichlet boundary conditions we found a class of exact solutions. Next, we considered the Neumann problem and showed that it reduces to a second order linear ordinary dierential equation and the standard heat equation with initial and nonlinear boundary conditions. Finally, we formulated the Cauchy problem for the variable parametric Burgers equation on the non-characteristic line, and obtained its solution in terms of a linear ODE and the series solution of the corresponding Cauchy problem for the heat equation. We gave examples to illustrate how some well known solutions of the Burgers equation can be recovered by solving a corresponding Cauchy problem.Master Thesis Integrable Vortex Dynamics and Complex Burgers' Equation(Izmir Institute of Technology, 2005) Gürkan, Zeynep Nilhan; Pashaev, Oktay; Gürkan, Zeynep Nilhan; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIntegrable dynamical models of the point magnetic vortex interactions in the plane are studied. Reformulating the Euler equations for vorticity in the Helmholtz form, the Hamiltonian and Lax representations are found. Reduction of these equations for the point vortices to the Kirchho equations, and non-integrability of the system of N 4 hydrodynamical vortices are discussed.As an integrable model of planar motion with given vorticity for the stationary and its solutions are given. For non-stationary planar vortex diffusion and exactly solvable Initial Value Problem for the one dimensional Burgers equation are solved.By the complexied Cole-Hopf transformation, the complex Burgers equation with integrable N vortex dynamics is introduced and linearization of this equation in terms of the complex Schr odinger equation is found.This allows us to construct N vortex congurations in terms of the complex Hermite polynomials, the vortex chain lattices and study their mutual dynamics. Mapping of our vortex problem to N-particle problem, the complexied Calogero-Moser system, showing its integrability and Hamiltonian structure is given. As an applicaton of the general results, we consider the problem of magnetic vortices in a magnetic fluid model. The holomorphic reduction of topological magnetic system to the linear complex Schrodinger equation, allows us to apply all results on integrable vortex dynamics in the complex Burgers equation to the magnetic vortex evolution, including magnetic vortex lattices and the bound states of vortices.