Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Resonance Solitons and Direct Methods in Soliton Theory(Izmir Institute of Technology, 2009) Duruk, Selin; Pashaev, Oktay; Pashaev, OktayThe Long-Short Wave interaction equations with adding quantum potential term and the Davey-Stewartson equation with addition of both, the quantum potential and the Hamiltonian terms are studied. These equations are reduced to different cases according to the choice of the quantum potential strength. For over critical case reductions to the non-linear diffusion-antidiffusion systems are derived. By the Hirota Direct Method one dissipaton solution of the system is derived. Two and three dissipaton (soliton) solutions are constructed explicitly. For special choice of the parameters they show the resonance character of interaction by fusion and fission of solitons.Master Thesis Nonlinear Euler Poisson Darboux Equations Exactly Solvable in Multidimensions(Izmir Institute of Technology, 2008) Ateş, Barış; Pashaev, OktayThe method of spherical means is the well known and elegant method of solving initial value problems for multidimensional PDE. By this method the problem reduced to the 1+1 dimensional one, which can be solved easily. But this method is restricted by only linear PDE and can not be applied to the nonlinear PDE. In the present thesis we study properties of the spherical means and nonlinear PDE for them. First we briefly review the main definitions and applications of the spherical means for the linear heat and the wave equations. Then we study operator representation for the spherical means, especially in two and three dimensional spaces. We find that the spherical means in complex space are determined by modified exponential function. We study properties of these functions and several applications to the heat equation with variable diffusion coefficient.Then nonlinear wave equations in the form of the Liouville equation, the Sine-Gordon equation and the hyperbolic Sinh-Gordon equations in odd space dimensions are introduced. By some combinations of functions we show that models are reducible to the 1+1 dimensional one on the half line.The Backlund transformations and exact particular solutions in the form of progressive waves are constructed. Then the initial value problem for the nonlinear Burgers equation and the Liouville equations are solved. Application of our solutions to spherical symmetric multidimensional problems is discussed.Master Thesis Hydrodynamic Interaction in Rotational Flow(Izmir Institute of Technology, 2007) Çağatay, Filiz; Yılmaz, OğuzThe interaction of water waves with arrays of vertical cylinders problem is studied using diffraction of water waves and addition theorem for bessel functions. The linear boundary value problem which is derived from physical assumptions is used as the approximate mathematical model for time-harmonic waves. Linearization procedure is described for the nonlinear boundary conditions on the free surface. The problem is solved by using Addition theorem for Bessel functions. Limiting case, k 0, known as long wave approximation, is analysed using limiting forms of Bessel functions. Vortex-cylinder interaction is analyzed using a similar technique involving Laurent series expansions of complex velocity and the Circle Theorem. But this method failed to work. Further analysis is necessary. Vortex dynamics is analysed in annular domains, which can conformally be mapped into infinite domain with two cylinders, using the terminology of q-calculus. Finally, the result of vortex-cylinder interaction in annular domain is transformed into the infinite domain with two cylinders using conformal mapping. Image representation clearly shows the mechanism of inverse images which accumulate at zero and infinity in the w-plane and a and 1/a in the z-plane.