Master Degree / Yüksek Lisans Tezleri

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

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Department: search.filters.department.Thesis (Master)--İzmir Institute of Technology, MathematicsPublisher: search.filters.publisher.Izmir Institute of Technology

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Now showing 1 - 10 of 74
  • Master Thesis
    A Comparison of Meshless and Finite Difference Methods for the Brusselator Model
    (Izmir Institute of Technology, 2022) Tanoğlu, Gamze; Tanoğlu, Gamze; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    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; Öz, Murat; Tanoğlu, Gamze; Tanoğlu, Gamze; 01. Izmir Institute of Technology; 04.02. Department of Mathematics; 04. Faculty of Science
    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
    Semigroup Theory and Some Applications
    (Izmir Institute of Technology, 2020) Batal, Ahmet; Batal, Ahmet; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    n the present thesis, we consider the evolution equation (Cauchy problem) which is the basis for our study. We show how various linear partial differential equations can be transformed into the Cauchy problem form. Solving the Cauchy problem is equivalent to find a family of evolution operators T(t) which sends the initial state of the system to the solution state at a later time t. It turns out that this family of operators T(t) must satisfy some properties which we call semigroup properties. We state the Hille-Yosida and Lumer-Phillips theorems to characterize contraction semigroups. Moreover, we apply these theorems to the heat and wave equations as examples. We also consider strongly continuous operator groups and Stone's theorem. Finally, we give some essential conditions to obtain wellposed evaluation equation and introduce an inhomogeneous Cauchy problem.
  • Master Thesis
    Algebraic Methods and Exact Solutions of Quantum Parametric Oscillators
    (Izmir Institute of Technology, 2019) Çetindaş, Osman; Atılgan Büyükaşık, Şirin; Pashaev, Oktay; Atılgan Büyükaşık, Şirin; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    In this thesis, we study different approaches for solving the Schrödinger equation for quantum parametric oscillators. The Wei-Norman algebraic approach, the Lewis- Riesenfeld invariant approach, the Malkin-Manko-Trifonov approach are investigated. For each approach, the wave function solutions of the Schrödinger equation, the propagator and dynamical invariants are found and their relations with each other are shown. In the Wei-Norman Algebraic approach, for constructing wave functions, explicit form of evolution operator is obtained uniquely in terms of two linearly independent classical solutions of the corresponding classical equation of motion. In Lewis-Riesenfeld approach, quadratic invariants are found in terms of the solution of Ermakov-Pinney equation and using the eigenstates of these invariants, wave function solutions are constructed. Setting initial values for Ermakov-Pinney solution, results of Wei-Norman and Lewis- Riesenfeld approaches are compared, then this solution is expressed in terms of same two linearly independent classical solutions. In Malkin-Manko-Trifonov approach, linear invariants which are symmetry operators for the Schrödinger equation, are constructed in terms of complex-valued solutions of the classical equation. Using these invariants, quadratic invariants are constructed and their eigenstates are used to find wave function solutions. Moreover, initial values for complex solutions of classical equation of motion are posed, and comparison of the three approaches is given.
  • Master Thesis
    On Generalization of Hopfian Modules
    (Izmir Institute of Technology, 2018) Yaman, Mehmet; Büyükaşık, Engin; Büyükaşık, Engin; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    The notion of Hopfian modules are defined as a generalization of modules of finite length as the modules whose surjective endomorphisms are isomorphisms. These modules and several generalizations of them are extensively studied in the literature. The aim of this thesis is to review some known results and extends some results about generalized Hopfian and weakly Hopfian modules. It is shown that a module is Hopfian if and only if it is both generalized Hopfian and weakly Hopfian. Torsion-free abelian groups are weakly Hopfian. Any nonsingular uniform module is weakly Hopfian. Direct summands of weakly Hopfian modules is weakly Hopfian. It is shown that direct sum weak Hopfian modules is not necessarily weakly Hopfian.
  • Master Thesis
    Kaleidoscope of Quantum Coherent States and Units of Quantum Information
    (Izmir Institute of Technology, 2018) Koçak, Aygül; Pashaev, Oktay; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    In the present thesis, we study superposition of coherent states as the kaleidoscope of quantum coherent states, associated with regular n-polygon symmetry and the roots of unity q2n = 1. These states are generalizations of the Schrödinger cat states, corresponding to the roots of unity q2 = −1. To describe physical characteristics of kaleidoscope states, we introduce new type of mod n exponential functions as a superposition of exponential functions in the form of discrete Fourier transform. These functions are also known as generalized hyperbolic functions, satisfying ordinary differential equations with proper initial conditions. Kaleidoscope states are eigenstates of n-th order eigenvalue problem for annihilation operator and are not minimal uncertainty states. These states are described as quantum Fourier transform of Glauber coherent states. Normalization factors, uncertainty relations, average number of photons and coordinate representation for these states are found in a compact form by mod n exponential functions. The set of kaleidoscope states, as orthonormal computatitonal basis of quantum states, describes generic qudit unit of quantum information. Relations of kaleidoscope states with quantum group symmetry are discussed. The special cases of trinity and quartet states, corresponding to qutrit and ququat units of quantum information are treated in details.
  • Master Thesis
    Discrete Fractional Integral Operators and Their Relations To Number Theory
    (Izmir Institute of Technology, 2018) Sert, Ezgi; Temur, Faruk; Temur, Faruk; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    The aim of this thesis is to get estimates on discrete fractional integral operators by using number theory. These operators, starting with the studies of Arkipov and Oskolkov, have been investigated for a long time. Fourier analysis and topics related to it have been used in these studies. However, this study will put forward new results on these operators with the help of arithmetic.
  • Master Thesis
    Classical Time Splitting Approaches and Their Error Anlyses for Nonlinear Differential Equations
    (Izmir Institute of Technology, 2018) Hacısalihoğlu, Elif; Tanoğlu, Gamze; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    In 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
    On the Structure of Modules Characterized by Opposites of Injectivity
    (Izmir Institute of Technology, 2018) Altınay, Ferhat; Büyükaşık, Engin; Büyükaşık, Engin; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    In this thesis we consider some problems and also generalize some results related to indigent modules and subinjectivity domains. We prove that subinjectivity domain of any right module is closed under factor modules if and only if the ring is right hereditary. Indigent modules are the modules whose subinjectivity domain is as small as possible, namely the modules whose subinjectivity domain is exactly the class of injective modules. We give a complete characterization of indigent modules over commutative hereditary Noetherian rings. The commutative rings whose simple modules are injective or indigent are fully determined. The rings whose cyclic right modules are indigent are shown to be semisimple Artinian. We also give a characterization of t.i.b.s. modules over Dedekind domains.
  • Master Thesis
    Generalized Golden-Fibonacci Calculus and Applications
    (Izmir Institute of Technology, 2018) Özvatan, Merve; Pashaev, Oktay; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    In the present thesis the Golden-Fibonacci calculus is developed and several applications of this calculus are obtained. The calculus is based on the Golden derivative as a finite difference operator with Golden and Silver ratio bases, which allowed us to introduce Golden polynomials and Taylor expansion in terms of these polynomials. The Golden binomial and its expansion in terms of Fibonomial coefficients is derived. We proved that Golden binomials coincide with Carlitz’ characteristic polynomials. By Golden Fibonacci exponential functions and related entire functions, the Golden-heat and the Golden-wave equations are introduced and solved. By introducing higher order Golden Fibonacci derivatives, related with powers of golden ratio, we develop the higher order Golden Fibonacci calculus. The higher order Fibonacci numbers, higher Golden periodic functions and higher Fibonomials appear as ingredients of this calculus. By using Golden Fibonacci exponential function, we introduce the generating function for new type of polynomials, the Bernoulli-Fibonacci polynomials and study their properties. As a geometrical application, the Apollonious type gaskets are described in terms of Fibonacci, Lucas and generalized Fibonacci numbers. Some mod 5 congruencies associated with Fibonacci and Lucas numbers are obtained.