http://dspace.cityu.edu.hk:80/handle/2031/774 2013-06-14T01:34:30Z 2013-06-14T01:34:30Z Wang, Hailing (汪海玲) http://dspace.cityu.edu.hk:80/handle/2031/6973 2013-06-13T02:37:49Z 2012-01-01T00:00:00Z

Title: Analytical solutions and bifurcation of nonlinear oscillators with discontinuities and impulsive systems by a perturbation-incremental method Authors: Wang, Hailing (汪海玲) Abstract: Nonlinear equations have been widely used in many areas of physics and engineering. They are of significant importance in mechanical and structural dynamics for the comprehensive understanding and accurate prediction of motion. Analytical solution obtained from classical perturbation methods such as Lindstedt-Poincar'e method, Krylov-Bogoliubov-Mitropolsky method, method of multiple scales and averaging method are usually accurate for small perturbation. For nonlinear oscillators with discontinuities, accurate analytical solution may not be easily obtained due to the nonsmooth property at the switching points. For impulsive systems, there is a sudden jump in the phase portrait. To the best of our knowledge, no harmonic balance method has ever been applied to investigate the bifurcation and continuation of period solutions of such systems. In this thesis, we investigate analytical solutions of nonlinear oscillators with discontinuities using a nonlinear time transformation method, and bifurcation and continuation of impulsive systems using a perturbation-incremental method. First, we study analytical periodic solutions of a generalized Duffing-harmonic oscillator having a rational form for the potential energy by a nonlinear time transformation method. An analytical solution is expressed in Pad'e approximation which often gives a better approximation of a function than its truncating Taylor series. Period solutions with large amplitude and those near to homoclinic/heteroclinic orbits are computed. Excellent agreement of the approximate presentations with the numerical simulation has been demonstrated and discussed. We also compared the results with those from the cubication method. Next, we present a nonlinear time transformation method to obtain analytical solutions of nonlinear oscillators with discontinuities. The essence of this method is that a periodic solution is approximated by the Chebyshev polynomials with a nonlinear time s rather than the physical time t. Since the first derivative of an approximate limit cycle oscillation obtained from the present method is piecewise continuous which agrees qualitatively with the exact solution, it gives accurate analytical solutions for the nonlinear oscillators with discontinuities. In some cases, the present method gives exact solutions while other perturbation methods give only approximate solutions. For those systems where exact solution is impossible, the approximate solution obtained from the present method is compared to He's homotopy perturbation method which is a powerful method with good accuracy for many systems. Finally, a perturbation-incremental (PI) method is presented for the bifurcation analysis of periodic solutions of impulsive systems. For such systems, a periodic solution is also approximated by the Chebyshev polynomials instead of the Fourier series so as to overcome the sudden jump in the phase portrait. In the perturbation step, a perturbed solution is obtained at bifurcation through solving a system of low-dimensional linear equations and is taken as an initial guess for incremental iteration. Through an incremental process, period solutions can be calculated to any desired degree of accuracy and their stabilities can be determined by the Floquet theory. As the parameter varies, period-doubling solutions leading to chaos can be identified. Notes: CityU Call Number: QA867.5 .W36 2012; vi, 123 leaves : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [101]-121)

2012-01-01T00:00:00Z Li, Buyang (李步揚) http://dspace.cityu.edu.hk:80/handle/2031/6972 2013-06-13T02:37:46Z 2012-01-01T00:00:00Z

Title: Mathematical modelling, analysis and computation of some complex and nonlinear flow problems Authors: Li, Buyang (李步揚) Abstract: This thesis consists of two parts: (I) modelling, analysis and computation of sweat transport in textile media; (II) unconditional convergence and optimal error analysis of the Galerkin FEM for nonlinear parabolic equations. The first part of the thesis is concerned with heat and sweat transport in porous textile media, which can be viewed as a nonisothermal, multiphase and multicomponent flow with complex phases changes. We present a more precise formulation of the condensation/evaporation process with a truncated Hertz-Knudsen equation, which makes the model applicable in the general dry-wet case. We introduce a flux type boundary condition for the fiber absorption equation to describe the absorption process in a wet environment more precisely, while the previous models with a simple saturated condition may not be realistic. Numerical simulations are performed to compare with experimental data, with both finite difference methods and finite element methods. Several practical cases are simulated for clothing assemblies with the human thermoregulation system. Moreover, we provide optimal error estimates for an uncoupled finite difference method in one-dimensional space and a splitting finite element method in three-dimensional space. The error analysis relies on some interesting skills used in PDEs analysis and physical features in modelling. The physical process of heat and sweat transport is governed by a system of nonlinear, degenerate and strongly coupled parabolic equations in general. However, mathematical analysis for these models is very limited due to the lack of reasonable link between modelling in engineering and analysis in mathematics. We prove existence of weak solutions for the dynamic models with complex phase changes. The proof is based on the nature of gas convections in the mass equations and energy equation, with physically realistic assumptions. The analysis presented in this thesis may be applied to the multicomponent heat and mass transport models in many other areas, and it also provides a fundamental tool for theoretical analysis of numerical methods. The second part of the thesis is concerned with unconditional convergence and optimal error analysis of the Galerkin/mixed finite element method for nonlinear parabolic equations, with commonly-used linearized semi-implicit schemes for the time discretization. To illustrate our method, we study the time-dependent nonlinear Joule heating equations and the equations of incompressible miscible flow in porous media, respectively. Optimal L2 error estimates are obtained without any time step restriction, while all the previous works required certain conditions for the time stepsize. Theoretical analysis is based on more precise analysis of a corresponding time-discrete partial differential equations. The approach used in this paper is applicable for more general nonlinear evolution equations and many other linearized semi-implicit (or implicit) time discretizations.for which previous works often require certain restrictions on the time stepsize τ. Notes: CityU Call Number: QC173.4.P67 L5 2012; 2, 221 leaves : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [207]-221)

2012-01-01T00:00:00Z Lee, Kei Fung (李奇峰) http://dspace.cityu.edu.hk:80/handle/2031/6971 2013-06-13T02:37:44Z 2012-01-01T00:00:00Z

Title: Uniform asymptotic expansions of the Tricomi-Carlitz polynomials and the Modified Lommel polynomials Authors: Lee, Kei Fung (李奇峰) Abstract: In this thesis, we derive uniform asymptotic expansions of the Tricomi-Carlitz poly- nomials f(α)n(x) and the modified Lommel polynomials hn,ν(x), as n → ∞, valid for x in (0,∞). Since these two polynomials do not satisfy a second-order differential equation, the powerful tools developed for differential equations are not applicable. Our discussion is divided into three parts. In the first part, we derive directly from the three-term recurrence relation (n+1)f(α)n+1(x)−(n+α)xf(α)n(x)+f(α)n−1(x) = 0, an asymptotic expansion for f(α)n(x) which holds uniformly in regions containing the critical values x=±2/√ν, where ν=n+2α−1/2. This method is based on the turning-point theory for three-term recurrence introduced by Wang and Wong [Numer. Math. 91 (2002) and 94 (2003)]. In the second part, the expansion is derived by using the cubic transformation for the integral ∫cJ(s;t) exp[νϕ(s;t)] ds, where J(s;t) and ϕ(s;t) are analytic functions of s, t is a bounded real parameter and ϕ(s; t) have two saddle points s±(t) which coalesce as t tends to some real number t0. Then we apply the integration-by-part technique suggested by Bleistein. As an application, an asymptotic expansion for the zeros of the Tricomi-Carlitz polynomials is derived. The validity for bounded t can be extended to unbounded t by using a sequence of rational functions introduced by Olde Daalhuis and Temme. The expansion involves the Airy functions and their derivatives. Error bounds are also given for one-term and two-term approximations. We finally study a asymptotic expansion for the modified Lommel polynomials hn,ν(t/N) which holds uniformly in regions containing the critical values x=±1/N, where N=n+ν. This method is again based on the turning-point theory for three-term recurrence; their zeros are also derived. Notes: CityU Call Number: QA404.5 .L44 2012; iv, 148 leaves : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [144]-148)

2012-01-01T00:00:00Z Gao, Ming (高明) http://dspace.cityu.edu.hk:80/handle/2031/6970 2013-06-13T02:37:42Z 2011-01-01T00:00:00Z

Title: Portfolio management, stochastic volatility and credit derivatives : three important issues in quantitative finance Authors: Gao, Ming (高明) Abstract: During the financial crisis, most stock markets experienced large drawdown from the peak, volatilities in financial markets increased significantly, and the credit market became illiquid. This thesis consists of three parts which study three important topics related to the issues listed above. First, we propose a dynamic investment strategy which can do a good job to follow the market when the market soars, and can retain a part of the profit gained from the soaring market when the market experiences dramatic drawdown. We analyze the behavior of such an investment strategy and validate it by the empirical study. Secondly, we derive an analytic asymptotic formula for pricing European options in the fast mean-reverting stochastic volatility model. Approximations available in literatures failed to capture the behavior of the option prices when the current volatility is very large. Our new formula is in excellent agreement with the fully numerical solutions of option prices. Thirdly, we propose a pricing framework for credit derivatives in illiquid markets. In our framework, the default intensity, the position of the current portfolio, the trading size and the risk aversion of the investor are key inputs for pricing credit derivatives. One can determine a quote price for a trade at the current position, and determine the trading size for given market prices. Notes: CityU Call Number: HG106 .G36 2011; iv, 129 leaves : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2011.; Includes bibliographical references (leaves [124]-129)

2011-01-01T00:00:00Z