Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis A Comparison of Meshless and Finite Difference Methods for the Brusselator Model(Izmir Institute of Technology, 2022) Akbalık, Leyla; Tanoğlu, Gamze; Tanoğlu, GamzeThe purpose of this thesis is the Brusselator model, which is used to model the reaction-diffusion occurring in processes of a chemical such as the formation of turing patterns in the skin of an animal, enzymatic reaction, ozone formation through triple collision with atomic oxygen; using methods such as Meshless Method and Finite Difference Method in space discretization and Runge Kutta Method, and the Adaptive Runge Kutta Method in time discretization, to find the method that gives more accurate. It is also to estimate the degree of the Meshless Method using the Finite Difference Method.Master Thesis A Numerical Approach for Optimization of Curing Kinetics of Composite Material(Izmir Institute of Technology, 2021) Öz, Murat; Tanoğlu, GamzeIn this thesis, we introduced a new method which is called the GMN (Gamze, Murat, Neslişah) algorithm. GMN algorithm determines the pre-exponential and activation energy of the curing process. This algorithm include tanh fitting for the measured conversion values via least squares minimization technique and linear fitting for the kinetic parameters. Experimentally determined differential scanning calorimetry (DSC) data sets for an epoxy resin functionalized by single wall carbon nanotubes are used for the verification of the proposed method. In computational part, in order to denote the effectiveness of the new proposed method, the results are also compared with the methods reported in the literature. To sum up, we have shown that the GMN algorithm provides a good match with the experimental data for all kinetic parameters.Master Thesis Classical Time Splitting Approaches and Their Error Anlyses for Nonlinear Differential Equations(Izmir Institute of Technology, 2018) Hacısalihoğlu, Elif; Tanoğlu, GamzeIn this thesis, Lie - Trotter splitting, Strang - Marchuk splitting and symmetrically weighted sequential (SWS) splitting methods which are known as classical operator splitting methods are considered to find the numerical solution of the various ordinary differential equations (ODEs) and partial differential equations (PDEs). We also presented their error analyses in order to show advantages and disadvantages of these methods. Firstly, we considered simple linear and nonlinear ODE examples to motivate for the classical operator splitting methods. Then, two numerical examples which consist of a kinetic model of phage infection and the Newell - Whitehead - Segel (NWS) equation are studied. All these examples show that the operator splitting methods are a powerful technique with respect to the accuracy and robustness.Master Thesis Finite Element Based Stabilized Methods for Time Dependent Convection-Diffusion Equation and Their Analysis(Izmir Institute of Technology, 2016) Yılmaz, Kemal Cem; Tanoğlu, GamzeThis study is focused on a Fourier stability and accuracy analysis of the time integration algorithms using generalized trapezioidal family of methods of scalar unsteady convection–diffusion equation with periodic boundary conditions. The discretization in space dimension is performed by standard Galerkin finite element formulation for low Peclet numbers and stabilized finite element formulation for large Peclet numbers. The stability analysis is performed namely by von-Neumann stability analysis. Accuracy is measured in terms of damping errors and phase speed errors. The behaviour of these temporal errors of the particular time stepping algorithms, i.e. forward Euler, Crank-Nicolson and backward Euler methods are compared with each other. Particular attention is given to the stabilized finite element formulation, that is the case where we consider high Peclet numbers. For this case, it is concluded that the Crank-Nicolson time stepping represents a better approximate solution compared to the other time integrators on transport process of an initial wave profile. Finally, at the end of the study, we derive a stabilization parameter under a particular condition on Courant number, which provides the relative phase speed error being almost equivalent to its optimal level, that is, the waves with different Fourier modes propagate almost in the same speed. Theoretical results are confirmed by a number of numerical experiments.Master Thesis Two Numerical Approaches for Solving Nonlinear Stiff Differential Equations(Izmir Institute of Technology, 2014) İmamoğlu, Neslişah; Tanoğlu, GamzeThis thesis presents two different numerical methods to solve non-linear stiff differential equations. The first method is exponential integrator, its error bounds are derived for the specific differential equations. Error analysis of exponential integrators is studied based on the Frèchet differentiation and Sobolev space. We obtain the error bounds in Hs(R) norms under the certain assumptions. The second method is a new iterative linearizaton technique. For the second one, we first time applied to general Frèchet derivative as a linearization technique for the numerical solution of nonlinear partial differential equations. In computational part, in order to denote the effectiveness of the new proposed method, we compare our proposed method with the well-known techniques with respect to the errors.Master Thesis Convergence Analysis and Numerical Solutions of the Fisher's and Benjamin-Bono Equations by Operator Splitting Method(Izmir Institute of Technology, 2014) Zürnacı, Fatma; Tanoğlu, GamzeThis thesis is concerned with the operator splitting method for the Fisher’s and Benjamin-Bono-Mahony type equations. We showthat the correct convergence rates inHs(R) space for Lie- Trotter and Strang splitting method which are obtained for these equations. In the proofs, the new framework originally introduced in (Holden, Lubich, and Risebro, 2013) is used. Numerical quadratures and Peano Kernel theorem, which is followed by the differentiation in Banach space are discussed In addition, we discuss the Sobolev space Hs(R) and give several properties of this space. With the help of these subjects, we derive error bounds for the first and second order splitting methods. Finally, we numerically check the convergence rates for the time step ∆t.Master Thesis Numerical Solutions of the Reaction-Diffusion Equations by Exponential Integrators(Izmir Institute of Technology, 2014) Sofyalıoğlu, Melek; Tanoğlu, GamzeThis thesis presents the methods for solving stiff differential equations and the convergency analysis of exponential integrators, namely the exponential Euler method, exponential second order method, exponential midpoint method for evolution equation. It is also concentrated on how to combine exponential integrators with the interpolation polynomials to solve the problems which has discrete force. The discrete force is approximated by using the Newton divided difference interpolation polynomials. The new error bounds are derived. The performance of these new combinations are illustrated by applying to some well-known stiff problems. In computational part, themethods are applied to linear ODE systems and parabolic PDEs. Finally, numerical results are obtained by using MATLAB programming language.Master Thesis Comparison of Geometric Integrator Methods for Hamilton Systems(Izmir Institute of Technology, 2009) İneci, Pınar; Tanoğlu, GamzeGeometric numerical integration is relatively new area of numerical analysis The aim of a series numerical methods is to preserve some geometric properties of the flow of a differential equation such as symplecticity or reversibility In this thesis, we illustrate the effectiveness of geometric integration methods. For this purpose symplectic Euler method, adjoint of symplectic Euler method, midpoint rule, Störmer-Verlet method and higher order methods obtained by composition of midpoint or Störmer-Verlet method are considered as geometric integration methods. Whereas explicit Euler, implicit Euler, trapezoidal rule, classic Runge-Kutta methods are chosen as non-geometric integration methods. Both geometric and non-geometric integration methods are applied to the Kepler problem which has three conserved quantities: energy, angular momentum and the Runge-Lenz vector, in order to determine which those quantities are preserved better by these methods.Master Thesis Operator Splitting Methhods for Differential Equations(Izmir Institute of Technology, 2010) Yazıcı, Yeşim; Tanoğlu, GamzeIn this thesis, consistency and stability analysis of the traditional operator splitting methods are studied. We concentrate on how to improve the classical operator splitting methods via Zassenhaus product formula. In our approach, acceleration of the initial conditions and weighted polynomial ideas for each cases are individually handled and relevant algorithms are obtained. A new higher order operator splitting methods are proposed by the means of Zassenhaus product formula and rederive the consistency bound for traditional operator splitting methods. For unbounded operators, consistency analysis are proved by the C0-semigroup approach. We adapted the Von-Neumann stability analysis to operator splitting methods. General approach to use Von-Neumann stability analysis are discussed for the operator splitting methods. The proposed operator splitting methods and traditional operator splitting methods are applied to various ODE and PDE problems.Master Thesis Numerical Solution of Highly Oscillatory Differential Equations by Magnus Series Method(Izmir Institute of Technology, 2006) Kanat, Bengi; Tanoğlu, GamzeIn this study, the differential equation known as Lie-type equation where the solutions of the equation stay in the Lie-Group is considered. The solution of this equation can be represented as an infinite series whose terms consist of integrals and commutators, based on the Magnus Series. This expansion is used as a numerical geometrical integrator called Magnus Series Method, to solve this type of equations. This method which is also one of the Lie-Group methods, has slower error accumulation and more efficient computation results during the long time interval than classical numerical methods such as Runge-Kutta, since it preserves the qualitative features of the exact solutions. Several examples are considered including linear and nonlinear oscillatory problems to illustrate the efficiency of the method.