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Exactly Solvable Burgers Type Equations With Variable Coefficients and Moving Boundary Conditions
(01. Izmir Institute of Technology, 2022) Atılgan Büyükaşık, Şirin; Atılgan Büyükaşık, Şirin; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
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) Atılgan Büyükaşık, Şirin; Atılgan Büyükaşık, Şirin; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
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.

Phd Degree / Doktora

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