Phd Degree / Doktora
Permanent URI for this collectionhttps://hdl.handle.net/11147/2869
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Doctoral Thesis Analysis and Application of Linearization Technique for Nonlinear Problems(Izmir Institute of Technology, 2020) İmamoğlu Karabaş, Neslişah; Tanoğlu, GamzeThe purpose of this thesis is to investigate the implementation of linearization technique combining with the multiquadric radial basis function method to nonlinear problems which appears in engineering and physics. Presented linearization technique is formed by the Frechet derivatives and Newton Raphson method. This technique is applied to Burgers' equation, Coupled Burgers' equation and 2-D cubic nonlinear Schrödinger equation. From the numerical results of the problems, it is believed that this technique can be used to solve other nonlinear and system of nonlinear partial differential equations numerically.Doctoral Thesis Numerical Methods for Nonlocal Problems(Izmir Institute of Technology, 2018) Kaya, Adem; Tanoğlu, GamzeIn this thesis, numerical methods for nonlocal problems with local boundary conditions from the area of peridynamics are studied. The novel operators that satisfy local boundary conditions were proposed as an alternative to the original nonlocal problems which uses nonlocal boundaries. Peridynamic theory is reformulation of continuum mechanics by integral equations for which it has some advantages over traditional partial differential equations. In peridynamic theory, a point can interact with other points within a certain distance which is called horizon and indicated by the parameter δ. In this thesis, we are particularly interested in role of the parameter δ in numerical methods for the novel problems. More precisely, we aim to show its role in condition number, discretization error and convergence factor of multigrid method.Doctoral Thesis Convergence Analysis of Operator Splitting Methods for the Burgers-Huxley Equation(Izmir Institute of Technology, 2015) Çiçek, Yeşim; Tanoğlu, GamzeThe purpose of this thesis is to investigate the implementation of the two operator splitting methods; Lie-Trotter splitting and Strang splitting method applied to the Burgers- Huxley equation and prove their convergence rates in Hs(R), for s ≥ 1. The analyses are based on the properties of the Sobolev spaces. The Burgers-Huxley equation is deal with the two parts; linear and non-linear parts. The regularity results are shown by using the same technique in (Holden, Lubich and Risebro, 2013) for both parts. By combining these results with the numerical quadratures and the Peano Kernel theorem error bounds are derived for the first and second order splitting methods. In the computational part, the operator splitting methods are applied to the Burgers-Huxley equation. Finally, the convergence rates for the two splitting methods are checked numerically. These numerical results confirmed the theoretical results.Doctoral Thesis New Approaches for Solving Nonlinear Oscillation Problems(Izmir Institute of Technology, 2015) Korkut Uysal, Sıla Övgü; Tanoğlu, GamzeThis thesis proposes two different numerical methods for solving nonlinear oscillation problems which appear in engineering and physics. Thus, the study is conducted in two parts. The first part introduces and analyzes the new iterative splitting method. In the construction of this method I utilize both the iterative splitting process and nonlinear Magnus expansion. Due to the fact that the iterative splitting procedure is employed, the constructed method can also be considered as a kind of operator splitting method. The second part presents a new linearization technique, based on the Newton-Raphson method and the Fréchet derivatives, for oscillation systems. Duffing oscillator and damped oscillator are used for testing the methods, respectively. Moreover, the proposed iterative splitting method and the proposed linearization technique are applied to both Van-der Pol equation and cubic nonlinear Schrödinger equation. Although the examples considered are a small sample of nonlinear oscillation equations, it is believed that the methods are easily adapted to solve such problems numerically. ivDoctoral Thesis Operator Splitting Method for Parabolic Partial Differential Equations: Analyses and Applications(Izmir Institute of Technology, 2013) Gücüyenen, Nurcan; Tanoğlu, GamzeThis thesis presents the consistency, stability and convergence analysis of an operator splitting method, namely the iterative operator splitting method, using various approaches for parabolic partial differential equations. The idea of the method is based first on splitting the complex problems into simpler equations. Then, each sub-problem is combined with iterative schemes and efficiently solved with suitable integrators. The analyses are based on the type of the operators of the problems. When the operators are bounded, the consistency is proved in two ways: first from derived explicit local error bounds and the second using the Taylor series expansion after combining iterative schemes with midpoint rule. As for the unbounded operators, since the Taylor series expansion is no longer valid, the consistency is derived using C0 semigroup theory. The stability is presented by constructing stability functions for each iterative schemes when the operators are bounded. For the unbounded, two stability analyses are offered: first one uses the continuous Fourier transform and the second uses semigroup theory. Lax- Richtmyer equivalence theorem and Lady Windermere’s fan argument which combine the stability and consistency are proposed for the convergence. In the computational part, the method is applied to three linear parabolic PDEs and to Korteweg-de Vries equation. These three equations are capillary formation model in tumor angiogenesis, solute transport problem and heat equation. Finally, numerical results are presented to illustrate the high accuracy and efficiency of the method relative to other classical methods. These numerical results align with the obtained theoretical results.Doctoral Thesis Stabilized finite element methods for time dependent convection-diffusion equations(Izmir Institute of Technology, 2012) Baysal, Onur; Tanoğlu, GamzeIn this thesis, enriched finite element methods are presented for both steady and unsteady convection diffusion equations. For the unsteady case, we follow the method of lines approach that consists of first discretizing in space and then use some time integrator to solve the resulting system of ordinary differential equation. Discretization in time is performed by the generalized Euler finite difference scheme, while for the space discretization the streamline upwind Petrov-Galerkin (SUPG), the Residual free bubble (RFB), the more recent multiscale (MS) and specific combination of RFB with MS (MIX) methods are considered. To apply the RFB and the MS methods, the steady local problem, which is as complicated as the original steady equation, should be solved in each element. That requirement makes these methods quite expensive especially for two dimensional problems. In order to overcome that drawback the pseudo approximation techniques, which employ only a few nodes in each element, are used. Next, for the unsteady problem a proper adaptation recipe, including these approximations combined with the generalized Euler time discretization, is described. For piecewise linear finite element discretization on triangular grid, the SUPG method is used. Then we derive an efficient stability parameter by examining the relation of the RFB and the SUPG methods. Stability and convergence analysis of the SUPG method applied to the unsteady problem is obtained by extending the Burman’s analysis techniques for the pure convection problem. We also suggest a novel operator splitting strategy for the transport equations with nonlinear reaction term. As a result two subproblems are obtained. One of which we may apply using the SUPG stabilization while the other equation can be solved analytically. Lastly, numerical experiments are presented to illustrate the good performance of the method.