Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis An Application of the Finite Differences Method To a Dynamical Interface Problem(Izmir Institute of Technology, 2004) Ağıroğlu, İzzet Onur; Tanoğlu, GamzeA multiple-order-parameter model for Cu-Au system on a face cubic centered lattice was recently developed in the presence of anisotropy. In that model, three order parameters (non-conserved) and one concentration order parameter (conserved), which has been taken as a constant, were considered. Later on, the model has been extended, so that, concentration has been taken as a variable. It has been seen that two models were in a good agreement near critical temperature since the non-conserved order parameter behaves like a constant near critical temperature in both models. Thus, we extended the rst model to a dynamical diffuse interface model near critical temperature. After writing the free energy of the system in terms of the order parameters, minimizing the energy with respect to the order parameters and Langevin equation yield the non-linear system of parabolic equations. The finite differences method was implemented to solve this non-linear system of parabolic equations. The forward difference discretization was applied for the rst derivative of the solution with respect to time and centered difference discretization was applied for the second order derivative of the solution with respect to spatial variable. We obtained stability criteria and nd the error bound. The orientation dependence proles, variation of interfacial energy and the effect of the degree of the anisotropy on the width of the diffuse interface are simulated when the time evolves.Master Thesis Analysis of Finite Difference Methods for Convection-Diffusion Problem(Izmir Institute of Technology, 2004) Demirayak, Murat; Neslitürk, Ali İhsanWe consider finite difference methods for one dimensional convection diffusion problem. An error analysis shows that the solution of the upwind scheme is not uniformly convergent in the discrete maximum norm due to its behavior in the layer. Then, we introduce and analyze a numerical method, Il.inAllen-Southwell scheme, that is first-order uniformly convergent in the discrete maximum norm throughout the domain. Finally, we present numerical results that confirm theoretical findings.Master Thesis The Solution of Some Differential Equations by Nonstandard Finite Difference Method(Izmir Institute of Technology, 2005) Kıran Güçoğlu, Arzu; Tanoğlu, GamzeIn this thesis, the nonstandard finite difference method is applied to construct thenew finite difference equations for the first order nonlinear dynamic equation, second order singularly perturbed convection diffusion equation and nonlinear reaction diffusion partial differential equation The new discrete representation for the first order nonlinear dynamic equation allows us to obtain stable solutions for all step-sizes.For singularly perturbed convection diffusion equation, the error analysis reveals that the nonstandard finite difference representation gives the better results for any values of the perturbation parameters. Finally, the new discretization for the last equation is obtained.The lemma related to the positivity and boundedness conditions required for the nonstandard finite difference scheme is stated. Numerical simulations for all differential equarions are illustrated based on the parameters we considered.