Phd Degree / Doktora

Permanent URI for this collectionhttps://hdl.handle.net/11147/2869

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Stability Analysis of Nonlinear Dynamical Systems With Lévy Typeperturbations
(01. Izmir Institute of Technology, 2023) Tamcı, Ege; Batal, Ahmet; Savacı, Ferit Acar
In order to model the noise in power networks, generally, normal distribution is used. However, normal distribution is not convenient in modelling noise which has sudden peaks. Instead, combination of a continuous process and a jump processes is much more suitable. With this idea in mind, in this thesis, the stability and control of two equations used in modeling power grids is analyzed, under the assumption that they are exposed to Lévy process noise which includes jumps. These equations are the swing equation and the Kuramoto Model. The swing equation is used to model the single machine infinite bus system (SMIBS). Kuramoto Model is used when a large number of generators are considered as a network of coupled oscillators with their own natural frequencies. In our stability control study in the SMIBS, the noise in the system has sudden and finite changes is assumed and therefore should be modelled with a modified tempered α-stable process obtained by adding a finite jump condition on the tempered α-stable process when α < 1. The control functions depending on the mechanical power input and damping parameters are designed in order to make the system stable in probability and exponential stable at its equilibrium point. These theoretical results are supported by numerical studies. For Kuromato model, assuming that the power network consists of two layers, namely oscillator, and control layers and that is affected with a general Lévy process which has finite jumps, functions which provide the stability of phase and frequencies are obtained, depending on oscillator and coupling strengths. In the light of the numerical studies, the control of frequency and phase synchronization up to a certain noise intensity level can be evaluated, but it is not possible beyond that level is concluded.
Boundary Feedback Stabilization of Some Evolutionary Partial Differential Equations
(01. Izmir Institute of Technology, 2022) Yılmaz, Kemal Cem; Batal, Ahmet; Özsarı, Türker; Özsarı, Türker; Batal, Ahmet
The purpose of this study is to control long time behaviour of solutions to some evolutionary partial differential equations posed on a finite interval by backstepping type controllers. At first we consider right endpoint feedback controller design problem for higher-order Schrödinger equation. The second problem is observer design problem, which has particular importance when measurement across the domain is not available. In this case, the sought after right endpoint control inputs involve state of the observer model. However, it is known that classical backstepping strategy fails for designing right endpoint controllers to higher order evolutionary equations. So regarding these controller and observer design problems, we modify the backstepping strategy in such a way that, the zero equilibrium to the associated closed-loop systems become exponentially stable. From the well-posedness point of view, this modification forces us to obtain a time-space regularity estimate which also requires to reveal some smoothing properties for some associated Cauchy problems and an initial-boundary value problem with inhomogeneous boundary conditions. As a third problem, we introduce a finite dimensional version of backstepping controller design for stabilizing infinite dimensional dissipative systems. More precisely, we design a boundary control input involving projection of the state onto a finite dimensional space, which is still capable of stabilizing zero equilibrium to the associated closed-loop system. Our approach is based on defining the backstepping transformation and introducing the associated target model in a novel way, which is inspired from the finite dimensional long time behaviour of dissipative systems. We apply our strategy in the case of reaction-diffusion equation. However, it serves only as a canonical example and our strategy can be applied to various kind of dissipative evolutionary PDEs and system of evolutionary PDEs. We also present several numerical simulations that support our theoretical results.
Exactly Solvable Burgers Type Equations With Variable Coefficients and Moving Boundary Conditions
(01. Izmir Institute of Technology, 2022) Bozacı, Aylin; Atılgan Büyükaşık, Şirin
In this thesis, firstly, a generalized diffusion type equation is considered. A family of analytical solutions to an initial value problem on the whole line for this equation is obtained in terms of solutions to the characteristic ordinary differential equation and the standard heat model by using Wei-Norman Lie algebraic approach for finding the evolution operator of the associated diffusion type equation. Then, initial-boundary value problems on half-line and an initial-boundary value problem with moving boundary for this equation are studied. It is shown that if the boundary propagates according to an associated classical equation of motion determined by the time-dependent parameters, then the analytical solution is obtained in terms of the heat problem on the half-line. For this, a non-linear Riccati type dynamical system, that simultaneously determines the solution of the diffusion type problem and the moving boundary is solved by a linearization procedure. The mean position of the solutions, the influence of the moving boundaries and the variable parameters are examined by constructing exactly solvable models. Then, an initial value problem for a generalized Burgers type equation on whole real line is discussed. By using Cole-Hopf linearization and solution of the corresponding generalized linear diffusion type equation, a family of analytical solution is obtained in terms of solutions to the characteristic equation and the standard heat or Burgers model. Exactly solvable models are constructed and the influence of the variable coefficients are examined. Later, an initial-boundary value problem for the generalized Burgers type equation with Dirichlet boundary condition defined on the half-line is studied. Finally, an initial-boundary value problem for the generalized Burgers type equations with Dirichlet boundary condition imposed at a moving boundary is considered. The analytical solution is obtained in terms of solution to characteristic equation and the standard heat or Burgers model, if the moving boundary propagates according to an associated classical equation of motion. In order to show certain aspects of the general results, some exactly solvable models are introduced and solutions corresponding to different types of initial and homogeneous/inhomogeneous boundary conditions are discussed by examining the influence of the moving boundaries.
Direct and Interior Inverse Generalized Impedance Problems for the Modified Helmholtz Equation
(01. Izmir Institute of Technology, 2022) Özdemir, Gazi; Ivanyshyn Yaman, Olha; Yılmaz, Oğuz
Our research is motivated by the classical inverse scattering problem to reconstruct impedance functions. This problem is ill-posed and nonlinear. This problem can be solved by Newton-type iterative and regularization methods. In the first part, we suggest numerical methods for resolving the generalized impedance boundary value problem for the modified Helmholtz equation. We follow some strategies to solve it. The strategies of the first method are founded on the idea that the problem can be reduced to the boundary integral equation with a hyper-singular kernel. While the strategy of the second approach makes use of the concept of numerical differentiation, the first approach treats the hyper singular integral operator by splitting off the singularity. We also show the convergence of the first method in the Sobolev sense and the solvability of the boundary integral equation. We give numerical examples which show exponential convergence for analytical data. In the second part of this work, we take into account the inverse scattering problem of reconstructing the cavity’s surface impedance from sources and measurements positioned on a curve within it. For the approximate solution of an ill-posed and nonlinear problem, we propose a direct and hybrid method which is a Newton-type method based on a boundary integral equation approach for the boundary value problem for the modified Helmholtz equation. As a consequence of this, the numerical algorithm combines the benefits of direct and iterative schemes and has the same level of accuracy as a Newton-type method while not requiring an initial guess. The results are confirmed by numerical examples which show that the numerical method is feasible and effective.
Exactly Solvable Quantum Parametric Oscillators in Higher Dimensions
(Izmir Institute of Technology, 2022) Çayiç, Zehra; Atılgan Büyükaşık, Şirin
The purpose of this thesis is to study the dynamics of the generalized quantum parametric oscillators in one and higher dimensions and present exactly solvable models. First, time-evolution of the nonclassical states for a one-dimensional quantum parametric oscillator corresponding to the most general quadratic Hamiltonian is found explicitly, and the squeezing properties of the wave packets are analyzed. Then, initial boundary value problems for the generalized quantum parametric oscillator with Dirichlet and Robin boundary conditions imposed at a moving boundary are introduced. Solutions corresponding to different types of initial data and homogeneous boundary conditions are found to examine the influence of the moving boundaries. Besides, an N-dimensional generalized quantum harmonic oscillator with time-dependent parameters is considered and its solution is obtained by using the evolution operator method. Exactly solvable quantum models are introduced and for each model, the squeezing and displacement properties of the time-evolved coherent states are studied. Finally, time-dependent Schrödinger equation describing a generalized two-dimensional quantum coupled parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. The propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, a Cauchy-Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields is introduced. Based on the explicit results, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.
Krull-Schmidt Properties Over Non-Noetherian Rings
(Izmir Institute of Technology, 2022) Gürbüz, Ezgi; Ay Saylam, Başak
Let R be a commutative ring and C a class of indecomposable R-modules. The Krull-Schmidt property holds for C if, whenever G1 ⊕ ·· · ⊕ Gn H1 ⊕ ·· · ⊕ Hm for Gi, Hj ∈ C, then n = m and, after reindexing, Gi Hi for all i ≤ n. The main purpose of this thesis is to investigate Krull-Schmidt properties of certain classes of modules over Non-Noetherian rings. Particularly weakly Matlis domains, strong Mori domains and Marot rings, all of which are among the class of Non-Noetherian rings, are studied. wweak isomorphism types are defined and the conditions when they coincide for torsionless modules over weakly Matlis domains are discussed. With the help of this comparison, the Krull-Schmidt property of w-ideals of a strong Mori domain is characterized. Also, the same property for overrings of a strong Mori domain is examined. Some useful results for a Marot ring with ascending condition on its regular ideals are obtained. Krull-Schmidt property on regular ideals of such a ring is studied and a characterization is given. Furthermore, the same property is discussed for overrings of a Marot ring.
When Certain Relative Projectivity and Injectivity Conditions Imply the Global Projectivity and Injectivity
(Izmir Institute of Technology, 2022) Benli Göral, Sinem; Büyükaşık, Engin
A right R-module M is called R-projective provided that it is projective relative to the right R-module RR. One of the parts of this thesis deals with the rings whose all nonsingular right modules are R-projective. For a right nonsingular ring R, we prove that RR is of finite Goldie rank and all nonsingular right R-modules are R-projective if and only if R is right finitely Σ-CS and flat right R-modules are R-projective. Then, R-projectivity of the class of nonsingular injective right modules is also considered. Over right nonsingular rings of finite right Goldie rank, it is shown that R-projectivity of nonsingular injective right modules is equivalent to R-projectivity of the injective hull E(RR). As a second goal, we deal with simple-injective modules. For a right module M, we prove that M is simple-injective if and only if M is min-N-injective for every cyclic right module N. The rings whose simple-injective right modules are injective are exactly the right Artinian rings. A right Noetherian ring is right Artinian if and only if every cyclic simple-injective right module is injective. The ring is quasi-Frobenius if and only if simple-injective right modules are projective. For a commutative Noetherian ring R, we prove that every finitely generated simple-injective R-module is projective if and only if R = A × B, where A is quasi-Frobenius and B is hereditary. An abelian group is simpleinjective if and only if its torsion part is injective.
Reidemeister torsion of closed л-manifolds
(Izmir Institute of Technology, 2021) Dirican Erdal, Esma; Sözen, Yaşar; Erman, Fatih
Let M be a closed orientable 2n-dimensional л-manifold such that n , 2 and M is either (n-2)-connected or (n-1)-connected. Such a manifold M can be decomposed as a connected sum of certain simpler manifolds. In this thesis, by using such connected sum decompositions, we develop multiplicative gluing formulas that express the Reidemeister torsion of M with untwisted R-coefficients in terms of Reidemeister torsions of its building blocks in the decomposition. Then we apply these results to handlebodies, compact orientable smooth (2n+1)-dimensional manifolds whose boundary is a (n-2)-connected 2n-dimensional closed л-manifold, and product manifolds.
Analysis and Application of Linearization Technique for Nonlinear Problems
(Izmir Institute of Technology, 2020) İmamoğlu Karabaş, Neslişah; Tanoğlu, Gamze
The purpose of this thesis is to investigate the implementation of linearization technique combining with the multiquadric radial basis function method to nonlinear problems which appears in engineering and physics. Presented linearization technique is formed by the Frechet derivatives and Newton Raphson method. This technique is applied to Burgers' equation, Coupled Burgers' equation and 2-D cubic nonlinear Schrödinger equation. From the numerical results of the problems, it is believed that this technique can be used to solve other nonlinear and system of nonlinear partial differential equations numerically.
Asymptotic Behaviour of Gravity Driven Free Surface Flows Resulting From Cavity Collapse
(Izmir Institute of Technology, 2020) Fetahu, Elona; Yılmaz, Oğuz
In this thesis, the gravity driven potential flows that result from cavity collapse are studied. Initially, the collapse of a vertical cylindrical cavity of circular cross sections surrounded by a liquid region is examined for two different situations. In the first one the cavity has same depth as the fluid and in the second one the cavity starts from the free surface and has less depth than the fluid. The problem is formulated by using a small parameter that represents the short duration of the stage. The first problem, as the radius and the centre of the cavity approach infinity, reduces to the classical two-dimensional dam break problem solved by Korobkin and Yilmaz (2009). The singularity of the radial velocity at the bottom circle is shown to be of logarithmic type. In the second problem, where the cavity is less deep than the fluid, the flow region is separated into two regions: the interior one, which is underneath the cylindrical cavity and above the rigid bottom, and the exterior one, which is the rest of the flow. The corresponding new problems are solved separately and then the coefficients are found by applying the matching conditions at the interface, where the fluid radial velocities and pressures coincide. On the limiting case, the problem reduces to the two-dimensional dam break flow of two immiscible fluids by Yilmaz et al. (2013a). Singularity at the bottom circle of the cavity is observed, which is of the same type as in the latter paper. Next, a third problem studies the gravity driven flow caused by the collapse of a rectangular section of a vertical plate. During the early stage, the flow is described by the velocity potential. Attention is paid to determining the velocity potential and free surface shapes. The solution follows the Fourier series method in Renzi and Dias (2013) and the boundary element method in Yilmaz et al. (2013a). Singularity is observed at the side edges and lower edge of the rectangular section. The horizontal velocity of the initially vertical free surface along the vertical line of symmetry of the rectangle is the same to the one in the two-dimensional problem Korobkin and Yilmaz (2009). The singularities observed in these problems lead to the jet formation for the initial stage. The methods applied in these computations are expected to be helpful in the analysis of gravity-driven flow free surface shapes. This thesis is a contribution towards the 3-D generalizations of dam break problems.

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