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Mon, 07 Apr 2014 01:15:25 GMT2014-04-07T01:15:25ZAnalytical solutions and bifurcation of nonlinear oscillators with discontinuities and impulsive systems by a perturbation-incremental method
http://dspace.cityu.edu.hk:80/handle/2031/6973
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)Sun, 01 Jan 2012 00:00:00 GMThttp://dspace.cityu.edu.hk:80/handle/2031/69732012-01-01T00:00:00ZMathematical modelling, analysis and computation of some complex and nonlinear flow problems
http://dspace.cityu.edu.hk:80/handle/2031/6972
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)Sun, 01 Jan 2012 00:00:00 GMThttp://dspace.cityu.edu.hk:80/handle/2031/69722012-01-01T00:00:00ZUniform asymptotic expansions of the Tricomi-Carlitz polynomials and the Modified Lommel polynomials
http://dspace.cityu.edu.hk:80/handle/2031/6971
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)Sun, 01 Jan 2012 00:00:00 GMThttp://dspace.cityu.edu.hk:80/handle/2031/69712012-01-01T00:00:00ZPortfolio management, stochastic volatility and credit derivatives : three important issues in quantitative finance
http://dspace.cityu.edu.hk:80/handle/2031/6970
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)Sat, 01 Jan 2011 00:00:00 GMThttp://dspace.cityu.edu.hk:80/handle/2031/69702011-01-01T00:00:00Z