Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
Browse
6 results
Search 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 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 Damping Oscillatory Models in General Theory of Relativity(Izmir Institute of Technology, 2007) Tığrak Ulaş, Esra; Pashaev, OktayIn my thesis we have studied the universe models as dynamical systems which can be represented by harmonic oscillators. For example, a harmonic oscillatior equation is constructed by the transformation of the Riccati differential equations for the anisotropic and homogeneous metric. The solution of the Friedman equations with the state equation satisfies both bosonic expansion and fermionic contraction in Friedman Robertson Walker universe with different curvatures is studied as a conservative system with the harmonic oscillator equations. Apart from the oscillator representations mentioned above (constructed from the universe models), we showed that the linearization of the Einstein field equations produces harmonic oscillator equation with constant frequency and the linearization of the metric on the de-Sitter background produces damped harmonic oscillator system. In addition to these, we have constructed the doublet and the Caldirola type oscillator equations with time dependent damping and frequency terms in the light of the Sturm Liouville form. The Lagrangian and Hamiltonian functions are calculated for all particular cases of the Sturm Liouville form. Finally, we have shown that zeros of the oscillator equations constructed from the particular cases can be transformed into pole singularities of the Riccati equations.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 Higher Order Symplectic Methods for Separable Hamiltonian Equations Master of Science(Izmir Institute of Technology, 2010) Gündüz, Hakan; Tanoğlu, GamzeThe higher order, geometric structure preserving numerical integrators based on the modified vector fields are used to construct discretizations of separable Hamiltonian systems. This new approach is called as modifying integrators. Modified vector fields can be used to construct high-order structure-preserving numerical integrators for both ordinary and partial differential equations. In this thesis, the modifying vector field idea is applied to Lobatto IIIA-IIIB methods for linear and nonlinear ODE problems. In addition, modified symplectic Euler method is applied to separable Hamiltonian PDEs. Stability and consistency analysis are also studied for these new higher order numerical methods. Von Neumann stability analysis is studied for linear and nonlinear PDEs by using modified symplectic Euler method. The proposed new numerical schemes were applied to the separable Hamiltonian systems.Master Thesis Uniformly Convergent Approximation on Special Meshes(Izmir Institute of Technology, 2007) Bingöl, Özgür; Neslitürk, Ali İhsanWe consider finite difference methods for the approximation of one-dimensional convection-diffusion problem with a small parameter multiplying the diffusion term. An analysis of the centered difference and upwind difference schemes on equidistant meshes shows that these methods are not uniformly convergent in the discrete maximum norm. However, we show that the upwind method over a set of suitably distributed mesh points produce uniformly convergent approximations in the discrete maximum norm. We further investigate the upwind difference method for the approximation of the convection-diffusion problem with a point source. Theoretical findings are supported with the numerical results.