Doktora Tezleri

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Analysis and application of linearization technique for nonlinear problems
(Izmir Institute of Technology, 2020-12) İmamoğlu Karabaş, Neslişah; Tanoğlu, Gamze; Izmir Institute of Technology
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, 2020-07) 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.
Boundary feedback stabilization of some evolutionary partial differential equations
(01. Izmir Institute of Technology, 2022-12) Yılmaz, Kemal Cem; 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.
Co-coatomically supplemented modules
(Izmir Institute of Technology, 2013-12) Güngör, Serpil; Büyükaşık, Engin
The purpose of this study to define co-coatomically supplemented modules, -cocoatomically supplemented modules, co-coatomically weak supplemented modules and co-coatomically amply supplemented modules and examine them over arbitrary rings and over commutative Noetherian rings, in particular over Dedekind domains. Motivated by cofinite submodule which is defined by R. Alizade, G. Bilhan and P. F. Smith, we define co-coatomic submodule. A proper submodule is called co-coatomic if the factor module by this submodule is coatomic. Then we define co-coatomically supplemented module. A module is called co-coatomically supplemented if every co-coatomic submodule has a supplement in this module. Over a discrete valuation ring, a module is co-coatomically supplemented if and only if the basic submodule of this module is coatomic. Over a non-local Dedekind domain, if a reduced module is co-coatomically amply supplemented then the factor module of this module by its torsion part is divisible and P-primary components of this module are bounded for each maximal ideal P. Conversely, over a non-local Dedekind domain, if the factor module of a reduced module by its torsion part is divisible and P-primary components of this module are bounded for each maximal ideal P, then this module is co-coatomically supplemented. A ring R is left perfect if and only if any direct sum of copies of the ring is -co-coatomically supplemented left R-module. Over a discrete valuation ring, co-coatomically weak supplemented and co-coatomically supplemented modules coincide. Over a Dedekind domain, if the torsion part of a module has a weak supplement in this module, then the module is co-coatomically weak supplemented if and only if the torsion part is co-coatomically weak supplemented and the factor module of the module by its torsion part is co-coatomically weak supplemented. Every left R-module is co-coatomically weak supplemented if and only if the ring R is left perfect.
Convergence analysis of operator splitting methods for the Burgers-Huxley equation
(Izmir Institute of Technology, 2015-07) Çiçek, Yeşim; Tanoğlu, Gamze
The purpose of this thesis is to investigate the implementation of the two operator splitting methods; Lie-Trotter splitting and Strang splitting method applied to the Burgers- Huxley equation and prove their convergence rates in Hs(R), for s ≥ 1. The analyses are based on the properties of the Sobolev spaces. The Burgers-Huxley equation is deal with the two parts; linear and non-linear parts. The regularity results are shown by using the same technique in (Holden, Lubich and Risebro, 2013) for both parts. By combining these results with the numerical quadratures and the Peano Kernel theorem error bounds are derived for the first and second order splitting methods. In the computational part, the operator splitting methods are applied to the Burgers-Huxley equation. Finally, the convergence rates for the two splitting methods are checked numerically. These numerical results confirmed the theoretical results.
Dinect and interior inverse generalized impedance problems for the modified Helmholtz equation
(01. Izmir Institute of Technology, 2022-11) 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.
Enriched finite elements method for convevtion-diffusion-reaction problems
(Izmir Institute of Technology, 2012) Şendur, Ali; Pashaev, Oktay
In this thesis, we consider stabilization techniques for linear convection-diffusionreaction (CDR) problems. The survey begins with two stabilization techniques: streamline upwind Petrov-Galerkin method (SUPG) and Residual-free bubbles method (RFB). We briefly recall the general ideas behind them, trying to underline their potentials and limitations. Next, we propose a stabilization technique for one-dimensional CDR problems based on the RFB method and particularly designed to treat the most interesting case of small diffusion. We replace the RFB functions by their cheap, yet efficient approximations which retain the same qualitative behavior. The approximate bubbles are computed on a suitable sub-grid, the choice of whose nodes are critical and determined by minimizing the residual of a local problem. The resulting numerical method has similar stability features with the RFB method for the whole range of problem parameters. We also note that the location of the sub-grid nodes suggested by the strategy herein coincides with the one described by Brezzi and his coworkers. Next, the approach in one-dimensional case is extended to two-dimensional CDR problems. Based on the numerical experiences gained with this work, the pseudo RFBs retain the stability features of RFBs for the whole range of problem parameters. Finally, a numerical scheme for one-dimensional time-dependent CDR problem is studied. A numerical approximation with the Crank-Nicolson operator for time and a recent method suggested by Neslitürk and his coworkers for the space discretization is constructed. Numerical results confirm the good performance of the method.
Entanglemend and topological soliton structures in Heisenberg spin models
(Izmir Institute of Technology, 2010) Gürkan, Zeynep Nilhan; Pashaev, Oktay
Quantum entanglement and topological soliton characteristics of spin models are studied. By identifying spin states with qubits as a unit of quantum information, quantum information characteristic as entanglement is considered in terms of concurrence. Eigenvalues, eigenstates, density matrix and concurrence of two qubit Hamiltonian of XY Z, pure DM, Ising, XY , XX, XXX and XXZ models with Dzialoshinskii- Moriya DM interaction are constructed. For time evolution of two qubit states, periodic and quasiperiodic evolution of entanglement are found. Entangled two qubit states with exchange interaction depending on distance J(R) between spins and influence of this distance on entanglement of the system are considered. Different exchange interactions in the form of Calogero- Moser type I, II, III and Herring-Flicker potential which applicable to interaction of Hydrogen molecule are used. For geometric quantum computations, the geometric (Berry) phase in a two qubit XX model under the DM interaction in an applied magnetic field is calculated. Classical topological spin model in continuum media under holomorphic reduction is studied and static N soliton and soliton lattice configurations are constructed. The holomorphic time dependent Schrödinger equation for description of evolution in Ishimori model is derived. The influence of harmonic potential and bound state of solitons are studied. Relation of integrable soliton dynamics with multi particle problem of Calogero-Moser type is established and N soliton and N soliton lattice motion are found. Special reduction of Abelian Chern-Simons theory to complex Burgers. hierarchy, the Galilean group, dynamical symmetry and Negative Burgers. hierarchy are found.
Exactly solvable Burgers type equations with variable coefficients and moving boundary conditions
(01. Izmir Institute of Technology, 2022-12) 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.
Exactly solvable quantum parametric oscillators in higher dimensions
(Izmir Institute of Technology, 2022-07) 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.
Homological objects of proper classes generated by simple modules
(Izmir Institute of Technology, 2014) Durğun, Yılmaz; Büyükaşık, Engin
The main purpose of this thesis is to study some classes of modules determined by neat, coneat and s-pure submodules. A right R-module M is called neat-flat (resp. coneat-flat) if the kernel of any epimorphism Y → M → 0 is neat (resp. coneat) in Y. A right R-module M is said to be absolutely s-pure if it is s-pure in every extension of it. If R is a commutative Noetherian ring, then R is C-ring if and only if coneat-flat modules are flat. A commutative ring R is perfect if and only if coneat-flat modules are projective. R is a right Σ -CS ring if and only if neat-flat right R-modules are projective. R is a right Kasch ring if and only if injective right R-modules are neat-flat if and only if the injective hull of every simple right R-module is neat-flat. If R is a right N-ring, then R is right Σ -CS ring if and only if pure-injective neat-flat right R-modules are projective if and only if absolutely s-pure left R-modules are injective and R is right perfect. A domain R is Dedekind if and only if absolutely s-pure modules are injective. It is proven that, for a commutative Noetherian ring R, (1) neat-flat modules are flat if and only if absolutely s-pure modules are absolutely pure if and only if R A × B, wherein A is QF-ring and B is hereditary; (2) neat-flat modules are absolutely s-pure if and only if absolutely s-pure modules are neat-flat if and only if R A × B, wherein A is QF-ring and B is Artinian with J2(B) = 0.
Krull-Schmidt properties over non-noetherian rings
(Izmir Institute of Technology, 2022-07) 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.
Mathematical modelling of light propagation in pohotonic crystal waveguides
(Izmir Institute of Technology, 2014) Eti, Neslihan; Sözüer, Hüseyin Sami
Photonic crystals are artificially engineered materials where the dielectric constant varies periodically. A photonic band gap can be created by scattering at the dielectric interfaces, which forbids propagation of light in a certain frequency range of light. This property enables us to control light, which is normally impossible with conventional optics. Moreover, by placing a linear defect into the photonic crystal, one can construct a waveguide, which keeps light inside the waveguide in the desired direction. Thus, by using photonic crystal waveguides one can control light propagation in integrated circuit devices. The goal of this work is to provide a comprehensive understanding of how to bend light using photonic crystal waveguides. The purpose is to create a 90◦ bend for line defect photonic crystal assisted waveguides and present fully three-dimensional calculations with optimized geometrical parameters that minimize the bending loss. The scheme uses one-dimensional photonic crystal slab waveguides for straight sections, and a corner element that employs a square photonic crystal with a band gap at the operating frequency.. The two different structures, with either silicon-silica or with silicon-air are used in the guiding photonic crystal layer. Furthermore, the guiding layer is sandwiched between either air on both top and bottom, or between air on top and silica substrate at the bottom, to serve as the ”cladding” medium. Calculations are presented for the transmission values of TE-like modes where the electric field is strongly transverse to the direction of propagation, with and without the photonic crystal corner element for comparison. We find that the bending loss can be reduced to under 2%.
Modules satisfying conditions that are opposites of absolute purity and flatness
(Izmir Institute of Technology, 2017-07) Kafkas Demirci, Gizem; Büyükaşık, Engin
The main purpose of this thesis is to study the properties which are opposites of absolute pure and flat modules. A right module M is said to be test for flatness by subpurity (for short, t.f.b.s.) if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. A left module M is said to be rugged if its flatness domain is the class of all regular right R-modules. Every ring has a t.f.b.s. module. For a right Noetherian ring R every simple right R-module is t.f.b.s. or absolutely pure if and only if R is a right V-ring or R A×B, where A is right Artinian with a unique non-injective simple right R-module and Soc(AA) is homogeneous and B is semisimple. A characterization of t.f.b.s. modules over commutative hereditary Noetherian rings is given. Rings all (cyclic) modules of whose are rugged are shown to be von Neumann regular rings. Over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S × T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical contains no properly nonzero ideals. Connections between rugged and poor modules are shown. Rugged Abelian groups are fully characterized.
Numerical methods for nonlocal problems
(Izmir Institute of Technology, 2018-07) Kaya, Adem; Tanoğlu, Gamze
In this thesis, numerical methods for nonlocal problems with local boundary conditions from the area of peridynamics are studied. The novel operators that satisfy local boundary conditions were proposed as an alternative to the original nonlocal problems which uses nonlocal boundaries. Peridynamic theory is reformulation of continuum mechanics by integral equations for which it has some advantages over traditional partial differential equations. In peridynamic theory, a point can interact with other points within a certain distance which is called horizon and indicated by the parameter δ. In this thesis, we are particularly interested in role of the parameter δ in numerical methods for the novel problems. More precisely, we aim to show its role in condition number, discretization error and convergence factor of multigrid method.
On relative projectivity of some classes of modules
(Izmir Institute of Technology, 2019-07) Alagöz, Yusuf; Büyükaşık, Engin
The main purpose of this thesis is to study R-projectivity and max-projectivity of some classes of modules, and module classes related to max-projective modules. A right R-module M is called max-projective provided that each homomorphism f:M → R/I where I is any maximal right ideal, factors through the canonical projection π:R → R/I. We call a ring R right almost-QF (resp. right max-QF) if every injective right R-module is R-projective (resp. max-projective). In this thesis we attempt to understand the class of right almost-QF (resp. right max-QF) rings. Among other results, we prove that a right Hereditary right Noetherian ring R is right almost-QF if and only if R is right max-QF if and only if R = S x T , where S is semisimple Artinian and T is right small. A right Hereditary ring is max-QF if and only if every injective simple right R-module is projective. Furthermore, a commutative Noetherian ring R is almost-QF if and only if R is max-QF if and only if R = A x B, where A is QF and B is a small ring. Moreover, we introduced and studied some homological objects related with max-projective modules.
Operations on proper classes related to supplements
(Izmir Institute of Technology, 2012) Demirci, Yılmaz Mehmet; Büyükaşık, Engin
The purpose of this study is to understand the properties of the operations +, ◦, and * defined on classes of short exact sequences and apply them to the proper classes related to supplements. The operation ◦ on classes of short exact sequences is introduced and it is proved that the class of extended weak supplements is the result of the operation ◦ applied to two classes one of which is the class of splitting short exact sequences. Using the direct sum of proper classes defined by R. Alizade, G. Bilhan and A. Pancar, a direct sum decomposition for quasi-splitting short exact sequences over the ring of integers is obtained. Closures of classes of short exact sequences along with the one studied by C. P. Walker, N. Hart and R. Alizade are defined over an integral domain. It is shown that these classes are proper when the underlying class is proper and they are related to the operation +. The closures of proper classes related to supplements are described in terms of Ivanov classes. Closures for modules over an integral domain are also defined and it is proved that submodules of torsion-free modules have unique closures. A closure for classes of short exact sequences is defined over an associative ring with identity and it is proved that this closure is proper when the underlying class is proper. Results shows that the operation + and closures of splitting short exact sequences plays an important role on the closures of proper classes.
Operator splitting method for parabolic partial differential equations: Analyses and applications
(Izmir Institute of Technology, 2013) Gücüyenen, Nurcan; Tanoğlu, Gamze
This thesis presents the consistency, stability and convergence analysis of an operator splitting method, namely the iterative operator splitting method, using various approaches for parabolic partial differential equations. The idea of the method is based first on splitting the complex problems into simpler equations. Then, each sub-problem is combined with iterative schemes and efficiently solved with suitable integrators. The analyses are based on the type of the operators of the problems. When the operators are bounded, the consistency is proved in two ways: first from derived explicit local error bounds and the second using the Taylor series expansion after combining iterative schemes with midpoint rule. As for the unbounded operators, since the Taylor series expansion is no longer valid, the consistency is derived using C0 semigroup theory. The stability is presented by constructing stability functions for each iterative schemes when the operators are bounded. For the unbounded, two stability analyses are offered: first one uses the continuous Fourier transform and the second uses semigroup theory. Lax- Richtmyer equivalence theorem and Lady Windermere’s fan argument which combine the stability and consistency are proposed for the convergence. In the computational part, the method is applied to three linear parabolic PDEs and to Korteweg-de Vries equation. These three equations are capillary formation model in tumor angiogenesis, solute transport problem and heat equation. Finally, numerical results are presented to illustrate the high accuracy and efficiency of the method relative to other classical methods. These numerical results align with the obtained theoretical results.
Quantum calculus of classical Heat-Burgers' hierarchy and quantum coherent states
(Izmir Institute of Technology, 2017-07) Nalcı Tümer, Şengül; Pashaev, Oktay
The purpose of this thesis is an application of quantum calculus to classical Heat- Burgers’ hierarchy and quantum coherent states. First we construct random walk on q-lattice, corresponding q-heat equation and exact solutions in terms of new family of q-exponential functions. Then we introduce a new type of q-diffusive heat equation and q-viscous Burgers’ equation, their polynomial solutions as generalized Kampe-de Feriet polynomials, corresponding dynamical symmetry and description in terms of Bell polynomials. Shock soliton solutions with fusion and fission of shocks are found and studied for different values of q. The q-semiclassical expansion of these equations in terms of Bernoulli polynomials is derived as corrections in power of ln q. A new class of complex valued function of complex argument as q-analytic functions in terms of q-analytic binomials is introduced and shown that these binomials are generalized analytic functions. As an application, we construct a new type of quantum states as q-analytic coherent states and corresponding q-analytic Fock- Bargmann representation. Then, we extend the concept of q-analytic function for two complex arguments, called double q-analytic functions, which has q-Hermite binomial expansion. As hyperbolic extension, we describe the q-analogue of traveling waves and find the D’Alembert solution of q-wave equation. By introducing q-translation operators we obtain q-binomials, q-analytic and q-anti analytic functions, q-travelling waves and non-commutative binomials. New type of quantum states as Hermite coherent states and Kampe-de Feriet coherent states are studied by generalization of the known Mehler formula. We introduce Golden quantum calculus, and as an application we study Golden quantum oscillator and its angular momentum representations.
Reidemeister torsion of closed л-manifolds
(Izmir Institute of Technology, 2021-07) Dirican Erdal, Esma; Sözen, Yaşar; Erman, Fatih; Izmir Institute of Technology
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.

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