Master Degree / Yüksek Lisans Tezleri

Permanent URI for this collectionhttps://hdl.handle.net/11147/3008

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COAR Access Rights: search.filters.coarAccessRights.open accessDepartment: search.filters.department.Thesis (Master)--İzmir Institute of Technology, Mathematics

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Now showing 1 - 10 of 84
  • 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, Gamze
    The 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
    On the Rings Whose Injective Modules Are Max-Projective
    (01. Izmir Institute of Technology, 2023) Yurtsever, Haydar Baran; Büyükaşık, Engin; Büyükaşık, Engin
    In this thesis, for some classes of rings including, local, semilocal right semihereditary and right Noetherian right nonsingular, we obtain some conditions that equivalent to being right max-QF. For example, for a semilocal right semihereditary ring, we prove that, the ring is right max-QF if and only if it is a direct product of a semisimple ring and a right small ring. A right Noetherian right nonsingular ring is right max-QF if and only if every injective module can be expressed as a direct sum of an injective module with no maximal submodules and a projective module. We show that, for a ring, being max-QF and almost-QF are not left-right symmetric. An example is given in order to show that max-QF and almost-QF rings are not closed under factor rings.
  • Master Thesis
    Fredholm Integral Equations of First Kind
    (01. Izmir Institute of Technology, 2023) Oruklu, Yıldız; Tanoğlu, Gamze; Ivanyshyn Yaman, Olha
    A 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, Gamze
    We 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, Gürhan
    In 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
    Qualitative Properties of Solutions of Some Keller-Segel Type Systems
    (01. Izmir Institute of Technology, 2022) Özdemir, Derya; Batal, Ahmet; Özsarı, Türker
    The main objective of this thesis is to summarize results related with solutions of some Keller - Segel type systems, which model chemotaxis. This work surveys mathematical studies starting with the work that first presented these systems in 1970. This study emphasizes the local and global existence of solutions of Keller - Segel type systems, in particular the boundedness and blow-up of solutions.
  • Master Thesis
    A Comparison of Meshless and Finite Difference Methods for the Brusselator Model
    (Izmir Institute of Technology, 2022) Akbalık, Leyla; Tanoğlu, Gamze; Tanoğlu, Gamze
    The 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, Gamze
    In 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
    Perturbative and Exact Analysis of Point Interactions
    (01. Izmir Institute of Technology, 2021) Altınbaşak, Ezgi; Erman, Fatih
    In this thesis, a general formulation for the bound state energies of a finite number of attractive Dirac delta potentials is given in terms of a finite dimensional matrix. The stationary scattering problem is also studied using the distributional solutions of algebraic equations in momentum space. Finally, the energy gap and splitting for the bound state energies when the distance between the delta potentials is large is approximately calculated using a kind of perturbation theory.
  • Master Thesis
    Wave Radiation From a Truncated Cylinder of Arbitrary Cross Sections
    (01. Izmir Institute of Technology, 2021) Korkmaz, Ece Hazal; Yılmaz, Oğuz
    Wave radiation problem in heaving motion from a vertical cylinder of circular cross-section and truncated cylinder of an arbitrary cross-section in the water of finite depth is studied. First, wave radiation from the circular cylinder is summarized which was solved analytically by Yeung (1981). The water domain is divided into two regions: the interior region below the cylinder and the exterior region outside the cylinder. The interior and exterior solutions are matched by the continuity of pressure and normal velocity in both cases. The vertical cylinder of a circular cross-section is solved by using the separation of variables method in cylindrical coordinates. The coefficients of interior and exterior solutions are related by the matching conditions. The system of equations formed by these unknown coefficients has been solved. Then, the non-dimensional z component of force is calculated by integrated pressure on the floating body. The real part and imaginary parts of this force give added mass and damping coefficients in heaving motion, respectively. These numerical results are used for the verification of asymptotic solutions of the present thesis. In the second case of this thesis, we treat wave radiation problems in heaving motion from the non-circular cylinder by using an asymptotic method. The asymptotic method of this thesis was suggested by Dişibüyük et al. (2017). Dişibüyük et al. (2017) suggested the non-dimensional maximum deviation of the cylinder cross-section from a circular one plays the role of a small parameter of the problem. The third-order asymptotic solution is used. Unknown coefficients of interior and exterior potentials are solved by using Fourier coefficients at each order of approximation. The advantage of the method is that the boundary conditions can be solved for different cross-sections by using the Fourier coefficients. The results are compared with other numerical results.