Master Degree / Yüksek Lisans Tezleri

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

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Author: search.filters.author.Tanoğlu, GamzePublisher: search.filters.publisher.01. Izmir Institute of Technology

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Now showing 1 - 4 of 4
  • 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; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    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
    Fredholm Integral Equations of First Kind
    (01. Izmir Institute of Technology, 2023) Tanoğlu, Gamze; Ivanyshyn Yaman, Olha; Tanoğlu, Gamze; Ivanyshyn Yaman, Olha; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    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) Tanoğlu, Gamze; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    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) Tanoğlu, Gamze; Tanoğlu, Gamze; Gürarslan, Gürhan; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    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.