Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Numerical Solution Methods for Boundary Value Problems for the Laplace Equation in Semi-Infinite Domains(01. Izmir Institute of Technology, 2023) Plattürk, Sabahat Defne; Tanoğlu, GamzeThe essential purpose of this thesis is to get numerical solutions of the Laplace Equation Boundary Value Problems subject to Dirichlet and Mixed boundary conditions on doubly connected semi-infinite domains, namely the upper half plane and semi-infinite strips, using boundary integral equations. Conformal maps served as a tool to transform the doubly connected semi-infinite domains into a doubly connected bounded domain. Images of boundary conditions are evaluated and the accuracy of the conformal maps are investigated. Then each problem is reduced to a system of linear boundary integral equations by representing the solution to the boundary value problems as combinations of double- and single-layer potentials. In the case of Dirichlet boundary conditions, we used a modification that ensures the unique solvability of the system of Fredholm Integral Equations of the second kind. However, in the case of mixed boundary conditions, such a modification is not needed. After the investigations of uniqueness and existence of solutions to the constructed systems of integral equations of the second kind, the systems of equations are solved by using the Nyström method, based on quadrature rules. For the numerical integration of integral operators with continuous kernels, the trapezoidal rule is used. For the numerical integration of the kernels with logarithmic singularity, we first split off the singularity and apply an extremely accurate quadrature rule for the improper integrals. Error analysis for both numerical integration techniques are given in details and the accuracy of Nyström Method which depend on the quadrature method is explained. Different test cases are considered to check the accuracy of the method and the order of convergence and error results are illustrated by numerical examples.Master Thesis Fredholm Integral Equations of First Kind(01. Izmir Institute of Technology, 2023) Oruklu, Yıldız; Tanoğlu, Gamze; Ivanyshyn Yaman, OlhaA unique variation of the inverse problem is the first type of Fredholm integral equation. To address the computing issue, inverse mathematical physics problems have been converted into the first type of Fredholm integral equation. We also use the Landweber iteration as an alternative to the well-known Tikhonov regularization technique , which has been shown to be most effective in solving ill-posed inverse problems. The Landweber iteration is a straight-forward and effective technique that exhibits convergence towards the accurate solution given specific conditions. Consequently, it serves as a valuable instru-ment for resolving inverse problems across diverse domains, including signal processing and geophysics. Following the examination of the properties of uniqueness and existence pertaining to solutions of integral equations of the first kind, the aforementioned equations are resolved through the utilization of the collocation method. The trapezoidal rule is widely utilized in numerical integration due to its straight-forward implementation and computational efficiency. However, it may not be appropriate for integrals with significant oscillatory behavior. In instances of this nature, it may be imperative to employ more sophisticated numerical integration methods, such as Gaussian quadrature or adaptive quadrature, in order to attain precise outcomes. For weakly singular integrals that appear in formulations of integral equations of potential problems in domains with corners and edges, we provide n-points Gaussian quadrature procedures which are particularly useful in numerical integration problems where the integral is difficult to evaluate. The accuracy of the method depends on the number of points used in the procedure, with higher order rules providing more accurate results.Master Thesis Legendre Wavelet Collocation Method With Quasilinearization Technique for Fractional Differential Equations(01. Izmir Institute of Technology, 2022) İdiz, Fatih; Tanoğlu, GamzeWe aim to present numerical methods based on Legendre wavelets and quasilinearization technique for fractional Lane-Emden type equations and time-fractional Fisher’s equation. The Lane-Emden equation is a second order singular non-linear ordinary differential equation, which is useful for modelling many astrophysical phenomena such as the distribution of stars in star clusters and star formation in molecular clouds. The Fisher’s equation is a non-linear reaction-diffusion equation that models the spread of mutant genes in a population. We start with a brief discussion of the purpose of studying fractional differential equations. Then some practical aspects of wavelets are explained. We also give introductory definitions and properties of fractional calculus and Legendre wavelets. Using Legendre wavelets and quasilinearization technique, we derive numerical methods for fractional Lane-Emden type equations and time-fractional Fisher’s equation. Moreover, the convergence analysis of both methods is studied. Some problems are solved to evaluate the efficiency of the proposed methods. Test problems show that the proposed methods are very effective.Master Thesis A Compact Finite Difference Method of Lines for Solving Non-Linear Partial Differential Equations(01. Izmir Institute of Technology, 2022) Ismoilov, Shodijon; Tanoğlu, Gamze; Gürarslan, GürhanIn this thesis, an efficient numerical method is proposed for the numerical solution of the chemical reaction-diffusion model governed by a non-linear system of partial differential equations known as a Brusselator model. The method proposed is based on a combination of higher-order Compact Finite Difference schemes and stable time integrator known as an adaptive step-size Runge-Kutta method. The performance of adaptive step-size Runge-Kutta formula of fifth-order accurate in time and Compact Finite Difference scheme of sixth-order in space are investigated. The method is implemented to solve three test problems and reveals that the method is capable of achieving high efficiency, accuracy and reliability. The results obtained are sufficiently accurate compared to some available results in the literature.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.