DSpace Collection:http://dspace.cityu.edu.hk:80/handle/2031/7752016-07-26T04:32:14Z2016-07-26T04:32:14ZNumerical solution of partial differential eigenvalue problems with the tau methodWong, Man (黃敏)http://dspace.cityu.edu.hk:80/handle/2031/77262015-09-25T02:01:10Z2012-01-01T00:00:00ZTitle: Numerical solution of partial differential eigenvalue problems with the tau method
Authors: Wong, Man (黃敏)
Abstract: The original idea of the Tau method is a numerical spectral method for approximating
the solution of linear ordinary differential equations (ODEs) which was proposed by
Lanczos [6] in 1938. Later, it was extended to the partial differential equations (PDEs)
with operation approach such that it can be easily implemented in the computer. The
method provided approximation in polynomial format which do not need initial guess,
fast convergences and small matrices manipulation are involved during the computation.
In recent decade, new development such as Tau–Line Method, Segmented Tau Method
and Tau–Collocation Method, etc. are introduced by many other researchers.
In this thesis, we will introduce Chebyshev polynomials which play an important
role in Tau method at the beginning. Then, we will demonstrate how to format the Tau
Problem from the differential equations such that we can find the Tau approximant to
the solution of the given problem. Also, we applied the Tau method into the operation
approach and the Tau–Collocation separately which reveal high performance and
accuracy method in numerical analysis of differential equations by some examples.
Consequently, we will use operation approach with Tau method to solve partial
differential eigenvalue problems. These problems are similar to equilibrium problems. It existed in many physical and engineering fields such as vibration of elastic material,
however, the major different is the eigenvalue problems involved the eigenvalues λ in
the system of equations. At last, we illustrated the technique by using the examples in
the Steklov's problems and the vibrations (also under variant tension) of clamped plates
which are eigenvalues λ existed in the boundary conditions and involving 4th orders
partial differential equation ∆2u − ∆u = λu respectively, Moreover, we will use Tau - Collocation Method to find the eigenvalues problems of PDE with variable coefficients.
Meanwhile, we will compare the result with eigenvalues found by other numerical
methods.
Notes: CityU Call Number: QA377 .W66 2012; 132 leaves : col. ill. 30 cm.; Thesis (M.Phil.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves 130-132)2012-01-01T00:00:00ZAnomalous phenomena and intermittency in the dynamics of multiple particles in a funnelGuo, Wenxuan ( 郭文軒)http://dspace.cityu.edu.hk:80/handle/2031/66332012-08-07T07:48:01Z2010-01-01T00:00:00ZTitle: Anomalous phenomena and intermittency in the dynamics of multiple particles in a funnel
Authors: Guo, Wenxuan ( 郭文軒)
Abstract: In this thesis, we investigate a granular system of multiple inelastic, frictionless, spherical
particles falling under gravity through a symmetric two-dimensional funnel with
different particle injection frequencies.
Intuitively, one might expect that the average durations which particles spend in
the funnel would increase monotonically as the particle injection frequency increases,
since, due to inter-particle collisions, the more particles in the funnel, the more likely
the particles would slow down and consequently more difficult to pass through the
funnel. However we show surprisingly that the duration is not a monotonic function of
the injection frequency. We provide an explanation for the seemingly counterintuitive
phenomena. We further conduct detailed studies on how the coefficient of restitution
and dynamics of inter-particle collisions affect the duration.
We also identify the phenomena of intermittency in the system with steep walls.
Namely, the behaviors of the system keep altering between two completely different
states at certain range of injection frequency. In one state, the particles have very few
inter-particle collisions, short durations and simple trajectories; in the other, they have
considerably more inter-particle collisions, much longer durations and more complicated
trajectories. We provide an explanation for these phenomena, and develop a
method of classifying these two states.
Notes: CityU Call Number: TA418.78 .G86 2010; iv, 36 leaves : ill. 30 cm.; Thesis (M.Phil.)--City University of Hong Kong, 2010.; Includes bibliographical references (leaves [35]-36)2010-01-01T00:00:00ZConvergence to shared lexicons for multi-object domainsXu, Chen (徐晨)http://dspace.cityu.edu.hk:80/handle/2031/62592011-05-25T01:19:35Z2010-01-01T00:00:00ZTitle: Convergence to shared lexicons for multi-object domains
Authors: Xu, Chen (徐晨)
Abstract: In recent years, consensus problems have attracted increasing attention from researchers
in various fields. Consensus problems in language emergence and evolution
are always raised in the following form: how might a group of agents reach a shared
communication system under certain patterns of interaction despite the absence of a
centralized coordinator?
In this thesis, consensus problems of a multi-agent model for multi-object domains
in discrete time are considered. We first propose a generalization of Liberman’s model
and study how a group of agents produce a common lexicon to describe the same collection
objects despite their different initial beliefs on word usage. At each time, all
agents meet together, select an object, exchange messages with a name for this object,
and update their beliefs, based on these messages, according to a designed protocol.
We study the dynamics for this model and analyze convergence and homonymy phenomena.
Then we study a different situation on which each agent can select its own individual
object. In contrast with the previous case, however, we now assume that agents can
exchange their beliefs (instead of only pairs (object, word)). We prove that all beliefs
will converge to the same belief provided each object is selected frequently enough.
Finally, computer simulations are attached to support the mathematical proofs of the
above two cases.
Notes: CityU Call Number: P326 .X8 2010; iv, 65 leaves : ill. 30 cm.; Thesis (M.Phil.)--City University of Hong Kong, 2010.; Includes bibliographical references (leaves [50]-52)2010-01-01T00:00:00ZOn a nonlinear model for stress-induced phase transitions in a slender compressible hyperelastic cylinder : analytical solutions and stabilityNg, Kwok-tim (伍國添)http://dspace.cityu.edu.hk:80/handle/2031/62572011-05-25T01:19:31Z2010-01-01T00:00:00ZTitle: On a nonlinear model for stress-induced phase transitions in a slender compressible hyperelastic cylinder : analytical solutions and stability
Authors: Ng, Kwok-tim (伍國添)
Abstract: In this thesis, some methodology in nonlinear dynamics is used to study a boundary-value problem of a nonlinear model arisen in phase transitions in a slender cylinder composed of a compressible hyperelastic material. We transform the original system of boundary value problem to an initial-value (dynamical) problem of finding periodic solutions of coupled nonlinear autonomous oscillators in a four-dimensional space. Hopf-like bifurcation analysis of the periodic solutions of the system is studied. Both analytical and numerical solutions are obtained by using a nonlinear transformation formulation. The analytical solutions are obtained by the perturbation method incorporate with a nonlinear transformation while the numerical solutions are obtained by the perturbation-incremental method. In addition, the accuracy of analytical solutions is investigated by comparing with the numerical solutions. The engineering stress-strain curve is plotted and compared with that from the normal form equation, which is a simplification of the original system. The stability of periodic solutions is also discussed in this thesis.
Notes: CityU Call Number: QA379 .N39 2010; iii, 99 leaves : ill. 30 cm.; Thesis (M.Phil.)--City University of Hong Kong, 2010.; Includes bibliographical references (leaves 78-80)2010-01-01T00:00:00Z