LEE- Matematik Mühendisliği-Yüksek Lisans
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ÖgeAnalysis and visualization of two and three dimensional data structures using a rational cubic spline function: A case study on water, natural gas, and electricity data in Istanbul(Graduate School, 2025-02-14)Spline is an interpolation technique that we use to best connect a series of data points in a mathematical sense and obtain a curved function. Basically, they consist of functions between every two data points, which we call multi-part polynomials, which are used to connect certain data points with a smooth curve. The function derivatives here are continuous. This means that the curve does not create corner breaks by making a smooth transition. Spline is also used to provide smooth transitions between data points, to create continuous curves without sharp corners or sudden changes, and to prevent excessive oscillations, i.e. unnecessary fluctuations, in higher-degree polynomial interpolations. Because in splines, instead of higher-degree polynomials, piecewise functions are used. It is preferred to use it when working with continuous data sets in data visualization, modeling data trends, engineering mathematics, computer graphics, numerical analysis, climate science, bioinformatics, machine learning and artificial intelligence. When we apply spline interpolation to real data sets, we obtain functions that provide smooth and continuous transitions between the data points we have. They are also used to estimate missing or noisy data, to determine trends or to model the dynamics of a system. There are many different spline methods in the literature. Different data sets and applications require the use of different spline types. The reason for choosing different spline types is the accuracy, calculation and shape control requirements that vary depending on the structure of the data set and modeling needs.In this thesis, different spline types are examined and linear spline, cubic spline and rational cubic spline methods known in the literature are used. While linear spline provides a fast and simple solution, cubic spline increases interpolation accuracy with smooth transitions. Rational cubic spline is advantageous in providing additional control conditions such as positivity. In this study, the rational cubic spline method has been examined in detail. There are many different forms of rational cubic spline. In this study, we use the form with four free parameters and C^1 continuity conditions. Shape control analyses have been performed to stretch the curve bends of these parameters or to provide control on the curve. The ability of shape control analyses to provide the desired shapes of parametric curves has been examined. In particular, in the analyses performed on rational cubic spline curves, it has been examined how well these curves fit the function. Rational cubic spline is a method that can preserve the positivity of the curve with the help of shape control parameters. Positivity is important in data that has a physical meaning, such as the data sets used here, such as water, natural gas and electricity consumption, which cannot be negative. In addition, three different derivative selection methods were used for rational cubic spline: Arithmetic method, geometric method and central difference method. Different derivative selections were evaluated in order to increase the accuracy of the curve and modeling success. While the arithmetic method provides an average transition, the geometric method captures proportional changes better. The central difference method increases the accuracy of interpolation with the symmetric derivative calculation approach. Peano Kernel Theorem is used to determine how close the rational cubic spline curve is to the function, to test the accuracy of spline derivatives, and to determine the upper bounds of the error in differential and integral error estimates. Peano Kernel Theorem is a method used in error analysis of spline interpolation and is used to compare and analyze the approximation errors of different spline types. As a result of these analyses, it is revealed which parameters should be optimized in what way in order to increase the precision of spline interpolation. The real datasets we used in this study are water, natural gas and electricity consumption data of Istanbul province. The datasets were obtained from the official website of Istanbul Metropolitan Municipality (IMM) and Istanbul Water and Sewerage Administration (ISKI). Python programming language was used in the visualizations. It is necessary to preserve positivity in rational cubic spline interpolation for the water, natural gas and electricity consumption datasets used in the study. Because water, natural gas and electricity consumptions can never be negative. They gain meaning when they are positive. For this reason, it is necessary to use a method that does not allow negative values and preserves positivity in the interpolation process. The spline interpolation methods used here, especially the rational cubic spline, were used to ensure that the consumption amounts in the dataset remain compatible with physical reality. Linear spline, cubic spline and rational cubic spline interpolation methods applied to real data sets provide a smooth and understandable representation of water, natural gas and electricity consumption data in time series format. Thanks to these methods, values that are not measured or are made but missing can be estimated. Precise analyses can be performed on consumption data. Since water, natural gas and electricity consumption quantities do not show a regular change over time, spline interpolations are sensitive to changes and fluctuations that may occur and can provide results accordingly. The use of these interpolation methods and visualization techniques provides valuable information for municipalities, energy and water distribution companies, environmental researchers and city planners. With these visualizations and models, the examination of consumption trends is very important in terms of efficient use of resources, establishing supply-demand balance and sustainability. In such cases, taking precautions against any possible situations is important in planning maintenance and repair processes and planning infrastructure investments. In two-dimensional visualizations, linear spline, cubic spline and rational cubic spline were used and derivatives for rational cubic spline were calculated with arithmetic, geometric and central difference methods. Each interpolation and derivative calculation method is colored so that users can compare the performance of different methods. Two-dimensional visualizations present the change of consumption data over time in a simple and understandable way, while allowing for clearer observation of increases or decreases in a certain period. In addition, error analyses were performed to evaluate the accuracy of interpolation methods and compare their performances. Five different interpolation metrics were used to perform these error analyses. These metrics were determined as mean absolute error MAE, mean square error MSE, coefficient of determination R^2, mean square logarithmic error RMSE and sum of squared error SSE. In this way, the accuracy and reliability levels of different methods on the data sets could be compared. Bicubic spline and rational bicubic spline interpolation methods were used for three-dimensional visualizations. In order to obtain three-dimensional images, the SciPy package in the Python programming language was used for cubic spline interpolation. For rational bicubic spline images, a rectangular region was defined and the rational cubic spline function was expanded. Through these visuals, water and natural gas consumption of 39 different districts of Istanbul province was presented in detail with changes on a yearly and monthly basis. Three-dimensional visualizations allow for a more holistic examination of temporal and spatial consumption patterns, allowing for more comprehensive analyses for both decision makers and researchers. Working with data sets and comparing various applications also helps evaluate consumption habits of different geographical, climatic and demographic characteristics. As a result, the use of such interpolation and visualization approaches supports faster, strategic, technical and effective decision-making in areas such as energy and water management, and contributes to the development of various policies for a more resilient infrastructure and resource management against possible problems in the future.
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ÖgeExact soliton solutions of cubic nonlinear Schrödinger equation with a momentum term(Graduate School, 2024-12-26)There are various interconnections between the positive sciences, with differential equations serving as a fundamental bridge linking mathematics to other scientific disciplines. Nonlinear wave phenomena have recently gained considerable attention due to their theoretical significance and applied relevance. Nonlinear optical wave equations not only facilitate the development of advanced techniques but also play a crucial role in elucidating natural phenomena across diverse fields, including biology, nonlinear optics, and quantum physics. Among these, solitons—localized nonlinear waves—stand out as valuable tools for understanding complex nonlinear systems. Solitons are widely studied in areas such as plasma physics, nonlinear optics, and quantum mechanics. Optical solitons, in particular, have drawn significant interest due to the inherently interdisciplinary nature of soliton theory, making it a pivotal topic for advancing technologies like high-speed data transmission. The external potential strongly influences the shape and stability of optical pulses. In quantum mechanics and nonlinear optics, potentials with parity-time symmetry (PT -symmetry) are frequently utilized. Numerous studies in the literature examine the stability of nonlinear Schrödinger (NLS) equations with PT -symmetry. These equations admit various nonlinear wave solutions, including solitons, which are localized waves that propagate without distortion. Solitons demonstrate remarkable resilience during collisions, retaining their properties even after interacting with other waves. This work investigates the soliton solutions and their stability in an NLS equation incorporating a momentum term and cubic nonlinearity under an external PT -symmetric potential. The governing equation is expressed as: iu_(z) +αu_(xx) −iΓu_(x) +φ|u|^2u+V_(PT) u = 0. Here, z denotes the scaled propagation distance, u is the differentiable complex-valued slowly varying amplitude, u_(xx) represents diffraction, Γ is the momentum term taken as a constant, and V_(PT) denotes the external potential. The PT -symmetric potential is defined as: VPT = V(x) +iW(x)=V0 +V1sech(x) +V2sech^2(x)+i[W0sech(x)tanh(x) +W1tanh(x)]. Here, V(x) and W(x) represent the real and imaginary components of the potential, where V(x) is an even function and W(x) is an odd function. A detailed introduction to solitons and their interdisciplinary significance is provided in Chapter 1. The NLS equation is introduced, along with its recent developments, including the momentum term and PT -symmetry. The chapter also outlines the research objectives and the thesis hypothesis, emphasizing the importance of the momentum term in the NLS equation. Chapter 2 describes the Spectral Renormalization (SR) Method, an iterative Fourier technique used to numerically solve the NLS equation with a momentum term and a PT -symmetric potential. The method is adapted to the problem at hand, and numerical solutions are obtained. In Chapter 3, the structure of the NLS equation without potential is analyzed to investigate the effect of the momentum term. Variations in soliton structures are examined in relation to changes in the momentum term coefficient, Γ, and the propagation constant, µ. Chapter 4 explores exact solutions of the NLS equation with a momentum term and PT -symmetric potential. Using the ansatz u(x,z) = f(x)e^i(µz+g(x)), where f(x) and g(x) are real-valued functions, analytical solutions are derived. These solutions are compared with the numerical results, which shows excellent agreement. The chapter also verifies the parity-time symmetry properties of the potential, confirming that its imaginary part is odd and its real part is even. Chapter 5 focuses on the stability analysis of soliton solutions. The Split-Step Fourier method is employed to investigate nonlinear stability, while linear stability is examined through the linear spectrum. The results indicate that the solitons become unstable with even slight increases in the momentum term coefficient, Γ. Additionally, enhancing the complex component of the potential increases instability, whereas increasing the real component improves stability. The acquired results are summed up in Chapter 6. Moreover, a brief discussion on potential future research is included. All numerical results were obtained using MATLAB2023®.
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ÖgeOn the hypersurfaces with non-diagonalizable shape operator in Minkowski spaces(Graduate School, 2022)Let $M$ be an oriented hypersurface of the Minkowski space $\mathbb E^{n+1}_1$. One of the most important extrinsic object of $M$ is its shape operator $S$ defined by the Wiengarten formula $$SX=-\tilde \nabla_X N,$$ where $N$ is the unit normal vector field to $M$ whenever $X \in TM$. The shape operator can be used to determine how the tangent plane and its normal move in all directions. Note that $S$ is a self adjoint endomorphism in $TM$. Therefore, it is diagonalizable when $M$ is Riemannian. However, if $M$ is Lorentzian, then its shape operator can be non-diagonalizable. In this case, the shape operator $S$ has four canonical forms. These canonical forms are written with respect to either an orthonormal basis or a pseudo-orthonormal basis. If the basis is orthonormal, then it is called a orthonormal frame field. An orthonormal frame of vector fields in a neighborhood of any point in $M$ is a basis $\{ E_1, \hdots, E_n \}$ such that $$(E_1,E_1)=-1, \quad (E_1,E_i)=0, \quad (E_i,E_j)=\delta_{ij}$$ for $2 \leq i, \ j \leq n$. On the other hand if the basis is pseudo-orthonormal, then it is called a pseudo-orthonormal frame field. A pseudo-orthonormal frame of vector fields in a neighborhood of any point in $M$ is a basis { X, Y, E_1, \hdots, E_{n-2} } such that $$(X,X)= (Y,Y)=0, \quad (X,E_i)=(Y,E_i)=0, \quad (X,Y)=-1$$ and $$(E_i, E_j)=\delta_{ij}$$ for $1 \leq i, \ j \leq n-2$. The eigenvalues and eigenvectors of $S$ are called the principal curvatures and principal directions of $M$, respectively. If the shape operator $S$ is diagonalizable and $M$ has constant principal curvatures, then the hypersurface $M$ is said to be isoparametric. If $S$ is non-diagonalizable and its minimal polynomial is constant, then $M$ is said to be isoparametric. In this thesis, we study isoparametric hypersurfaces with non-diagonalizable shape operator in Minkowski space $\mathbb E^{4}_1$. This thesis consists of five sections. First section is introduction. In the second section, we give some basic concepts on Lorentzian inner product and also some basic facts on hypersurfaces of $\mathbb E^{n+1}_1$. In the third section, a theorem about isoparametric hypersurfaces is given. We note that these theorems are proved by Magid in 1985. We prove these theorems by using another method. In fact, this theorem implies that there is only four classes of isoparametric hypersurface using the Codazzi and Gauss equations in $\mathbb E^{4}_1$. Then, we give another theorem which proves that there is no isoparametric hypersurface in $\mathbb E^{4}_1$ with complex principal curvatures. In the fourth section, we construct a family of hypersurfaces with non-diagonalizable shape operator in $\mathbb E^{5}_1$. We obtain the shape operator, the mean curvature, Gauss-Kronecker curvature and Levi-Civita connection of this hypersurface. Then, we give the necessary and sufficient condition for this hypersurface to be minimal with a theorem.
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ÖgeSuppression of symmetry-breaking bifurcations of optical solitons in parity-time symmetric potentials(Graduate School, 2022)Optical soliton refers to any optical field that maintains its special structure during propagation because of the balance between diffraction and self-phase modulation of the medium. The dynamics of optical solitons are investigated comprehensively due to their fundamental structures and potential applications. In particular, optical solitons play an important role in fiber optic communication system that uses pulses of infrared light to transmit information from one place to another over a long distance. The propagation of the electromagnetic wave in optical fibers is modeled by the cubic-quintic nonlinear Schrödinger (CQNLS) equation iΨ_z+Ψ_{xx}+α|Ψ|^2Ψ+β|Ψ|^4Ψ=0, where Ψ(x,z) is normalized complex-valued slowly varying pulse envelope of the electric field, z is the scaled propagation distance, x is the transverse coordinate, Ψ_{xx} corresponds to diffraction, α and β are the coefficients of cubic and quintic nonlinearities, respectively. A higher-order dispersion needs to be considered for performance enhancement along trans-oceanic and trans-continental distances. Fourth order dispersion needs to be taken into account for short pulse widths where the group velocity dispersion changes within the spectral bandwidth of the signal. In addition, it is known from many studies in the literature that an external potential added to the system can be also beneficial for performance improvement. In this thesis, we consider the nonlinear paraxial beam propagation in cubic-quintic nonlinearity with a complex parity-time (PT) symmetric potential and fourth order dispersion. This propagation is modeled by the following CQNLS equation iΨ_z+Ψ_{xx}−γΨ_{xxxx}+V(x)Ψ+α|Ψ|^2Ψ+β|Ψ|^4Ψ=0, where γ>0 is the coupling constant of the fourth order dispersion, V(x) represents a complex PT-symmetric potential. In this thesis, we consider PT-symmetric potentials that are of the form V(x)=g^2(x)+c0*g(x)+ig′(x) where g(x) is an arbitrary real and even function, c0 is an arbitrary real constant and PT-symmetric solitons undergo symmetry breaking. We take a localized double-hump function g(x) in the form of g(x)=A*[exp(−(x+x0)^2)+exp(−(x−x0)^2)] where A and x0 are related to the modulation strength and separation of PT-symmetric potential, respectively. The soliton solutions of CQNLS equation with fourth order dispersion and a complex PT-symmetric potential are numerically obtained by means of the squared-operator method since the equation is nonintegrable. The linear stability analysis of the numerically obtained solitons is examined by linear spectrum analysis and the nonlinear stability analysis is examined by nonlinear evolution with split-step Fourier method. The existence of symmetry breaking of solitons and suppression of symmetry-breaking bifurcations have been investigated. To examine the effect of fourth order dispersion on this symmetry breaking, the coefficient of fourth order dispersion γ is incremented gradually. Consequently, we have demonstrated that the symmetry-breaking bifurcation of the solitons in this problem is completely suppressed as the strength of the fourth order dispersion increases. Moreover, increasing the strength of fourth order dispersion positively influences the linear and nonlinear stability behaviors of solitons.
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ÖgeBiconservative and biharmonic surfaces in Euclid and Minkowski spaces(Graduate School, 2024-07-02)In 1964, Sampson and Eells formulated the concept of biharmonic maps as an extension of harmonic maps while investigating the energy functional $E$ between Riemannian manifolds, a subject of both geometric and physical significance. Subsequently, numerous mathematicians have shown interest in the study of biharmonic mappings.\\ By the definition, the bienergy functional between semi-Riemannian manifolds $(M^m,g)$ and $(N^n,\tilde{g})$ is defined by $$E_2(\varphi)=\frac {1}{2} \int_M \|\tau(\varphi)\|^2 v_g$$ for a smooth map $\varphi:M \to N$, where $\tau(\varphi)$ represents the tension field of $\varphi$. $\varphi:M\to N$ is said to be biconservative if it is a critical point of $E_2$. This condition is equivalent to satisfying the Euler-Lagrange equation associated with the bienergy functional $$\tau_2(\varphi)=0,$$ where $\tau_2$ is the bitension field defined by $$\tau_2(\varphi):=\Delta\tau(\varphi)-\mathrm{tr\,} \tilde{R}(d\varphi,\tau(\varphi))d\varphi.$$\\ In the 1980s, B. Y. Chen conducted research on biharmonic submanifolds within Euclidean spaces as a component of B. Y. Chen's initiative to comprehend submanifolds of finite type in semi-Euclidean spaces. B. Y. Chen proposed an other characterization of biharmonic submanifolds in these spaces. Let $x:M\to \mathbb E^n_r$ be an isometric immersion. By examining normal and tangential parts of $\tau_2(x)$, the following results can be obtained.\\ Let $x: M^m \rightarrow \mathbb E^n_r$ be an isometric immersion of an $n$-dimensional semi-Riemannian submanifold $M^m$ into the semi-Euclidean space $\mathbb E^n_r$. If $x$ satisfies the fourth-order semi-linear PDE system given by the equations $$\Delta^\perp H+\mathrm{trace}h(A_H(\cdot),\cdot)=0$$ and $$m\mathrm{grad} \Vert H \Vert^2 +4\mathrm{trace} A_{\nabla^\perp_\cdot H}(\cdot)=0,$$ then $M^m$ is biharmonic.\\ On the other hand, if a mapping $\varphi: M \to N$ satisfies the weaker condition $$\langle \tau_2(\varphi), d\varphi \rangle = 0,$$ then it is said to be biconservative. Mainly, if $x: M \to N$ is an isometric immersion, then the previous equation is equivalent to $$\tau_2(x)^\top = 0,$$ where $\tau_2(x)^\top$ represents the tangential part of $\tau_2(x)$. In this case, $M$ is said to be a biconservative submanifold of $N$.\\ In this thesis, we mainly focus on biharmonic and biconservative surfaces in four dimensional Euclidean and Minkowski spaces. The first section provides a concise overview of the historical background and underlying principles concerning biharmonic and biconservative submanifolds, as well as an overview of the research conducted far in this field. In the second section, we give some basic notations and basic facts about submanifolds of semi- Euclidean spaces, the definition of biconservative submanifolds and we introduce the rotational surfaces. In the third section, we give biconservative PNMCV surfaces in $\mathbb E^4$. We obtain local parameterizations of these surfaces and demonstrate that they are not biharmonic. In the fourth section, we give biconservative rotational surfaces in $\mathbb E_1^4$. We study with three different class of rotational surfaces and obtain the condition for each of them to be biconservative. In the concluding section, the derived conclusions are presented, along with recommendations regarding possible future researches.