by Ma Ka Chun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 50-52). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.7 / Chapter 2 --- Queueing Networks --- p.9 / Chapter 2.1 --- One-queue Networks --- p.9 / Chapter 2.2 --- Two-queue Free Networks --- p.12 / Chapter 2.3 --- Two-queue Overflow Networks --- p.13 / Chapter 2.4 --- Networks with Batch Arrivals --- p.14 / Chapter 3 --- ODE Solvers --- p.16 / Chapter 3.1 --- The Initial Value Methods --- p.16 / Chapter 3.1.1 --- The Linear System of Ordinary Differential Equations --- p.16 / Chapter 3.1.2 --- Euler's Method --- p.17 / Chapter 3.1.3 --- Runge-Kutta Methods --- p.17 / Chapter 3.1.4 --- The Stability of the IVMs --- p.19 / Chapter 3.1.5 --- Applications in Queueing Networks --- p.20 / Chapter 3.2 --- The Boundary Value Methods --- p.20 / Chapter 3.2.1 --- The Generalized Backward Differentiation For- mulae --- p.21 / Chapter 3.2.2 --- An example --- p.24 / Chapter 4 --- The Linear Equation Solver --- p.26 / Chapter 4.1 --- Iterative Methods --- p.26 / Chapter 4.1.1 --- The Jacobi method --- p.27 / Chapter 4.1.2 --- The Gauss-Seidel Method --- p.28 / Chapter 4.1.3 --- Other Iterative Methods --- p.29 / Chapter 4.1.4 --- Preconditioning --- p.29 / Chapter 4.2 --- The Multigrid Method --- p.30 / Chapter 4.2.1 --- Iterative Refinement --- p.30 / Chapter 4.2.2 --- Restriction and Prolongation --- p.30 / Chapter 4.2.3 --- The Geometric Multigrid Method --- p.33 / Chapter 4.2.4 --- The Algebraic Multigrid Method --- p.38 / Chapter 4.2.5 --- Higher Dimensional Cases --- p.38 / Chapter 4.2.6 --- Applications in Queueing Networks --- p.38 / Chapter 5 --- Numerical Experiments --- p.41 / Chapter 6 --- Concluding Remarks --- p.49 / Bibliography --- p.50

Firstly, we consider the following critical elliptic Neumann problem --Deltau + muu = uN+2N-2 , u > 0 in O; 6u6n = 0 on ∂O, where O is a smooth bounded domain in RN , N ≥ 7, mu is a large positive number and nu denotes exterior unit normal vector. We show that at a positive nondegenerate local minimum point Q0 of the mean curvature function, for any fixed integer K ≥ 2, there exists a mu K > 0 such that for mu > muK, the above problem has K -- bubble solution umu concentrating at the same point Q 0. Precisely, we show that umu has K local maximum points Qm1,...,Qm K ∈ ∂O with the property that umQmj ∼mN-22 ,Qmj→Q0 , j = 1, ..., K, and mN-3N Q'1 m,...,Q'K m approaches an optimal configuration that minimizes the following functional RQ'1,...,Q 'K=c1 j=1K4Q' j+c2 i≠j1&vbm0;Q' i-Q'j&vbm0;N-2 where Qmi=Qm i,1,...,Qmi,N-1 ,Qmi,N:= Q'i m,Qmi,N , c1, c2 > 0 are two generic constants and ϕ(Q) = Q T GQ with G = (∇ijH(Q0)). / In my thesis, I will address different concentration phenomena for some second order elliptic problems. / Lastly, we consider the problem &egr;2Delta u -- u + uq = 0 in a smooth bounded domain O ⊂ R2 with Neumann boundary condition where &egr; > 0 is a small parameter and q > 1. We prove for some explicit &egr;'s the existence of positive solution u&egr; concentrating at any connected component of ∂O, exponentially small in &egr; at any positive distance from it. / Secondly, we study positive solutions of the equation &egr;2Delta u -- u + uN+2N-2 = 0, where N = 3, 4, 5, and &egr; > 0 is small, with Neumann boundary condition in a smooth bounded domain O ⊂ RN . We prove that, along some sequence {&egr;j} with &egr;j → 0, there exists a solution with an interior bubble at an innermost part of the domain and a boundary layer on the boundary ∂O. / Wang, Liping. / "June 2008." / Adviser: Jun Cheng Wei. / Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1707. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 107-117). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

In [DS1,2,3], Denker and Sato initiated a new point of view to study the problem. They identified the Sierpinski gasket as a Martin boundary of some canonical Markov chain and used the associated theory to consider the problem. In this thesis, we will extend their result so as to be applicable to all single-point connected monocyclic post critically finite (m.p.c.f.) self-similar sets. / In the first chapter, we review some basic facts of the self-similar sets and the Martin boundaries, and we prove that every m.p.c.f. self-similar set K is homeomorphic to the quotient space of the symbolic space associated with K, moreover, the homeomorphism is a Lipschitz equivalence for some special m.p.c.f. self-similar sets. / In the second chapter, we first prove that the quotient space of the symbolic space associated with K is homeomorphic to the Martin boundary with respect to the state space associated with K if K is a single-point connected m.p.c.f. self-similar set. Combining this result and the result in the first chapter, we conclude that every single-point connected m.p.c.f. self-similar set can be identified with the Martin boundary of some canonical Markov chain. Then for the 3-level Sierpinski gasket, we prove that there exists a one to one relation between the strongly P-harmonic functions on the 3 state space and K-harmonic functions constructed by Kigami. / In the third chapter, we define a new Markov chain on the pentagasket K which is a single-point connected m.p.c.f. self-similar also. Under the new Markov chain, we prove that K can be identified with the Martin boundary of the new Markov chain and that there exists a one to one relation between the strongly P-harmonic functions and the K-harmonic functions. / One of the fundamental problems in fractal analysis is to construct a Laplacian on fractals. Since fractals, like the Sierpinski gasket and the pentagasket, do not have any smooth structures, it is not possible to construct it from the classical point of view. Hence, until now there is no systematic way to define such a notion on the general class of fractals. / There are two approaches for the problem which have achieved some success in certain special situations. The first one is a probabilistic approach via constructing Brownian motions on self-similar sets. The second approach is an analytical one proposed by Kigami. He approximated the underlying self-similar set K by an increasing sequence of finite sets equipped with the discrete Laplacians Hm in a consistent way. He showed that if K is strongly symmetric, then Hm converge to a Laplacian on K. / by Ju, Hongbing. / "March 2008." / Adviser: Lau Ka Sing. / Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1702. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 91-94). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

In chapter 1, we focus on the full potential equation in an infinite nozzle with some decay cross-sections and prove the existence and stability of the transonic shock wave; which is a solution to a free boundary value problem for a quasi-linear mix-typed partial differential equation with the position of shock as a free boundary. To achieve this conclusion, we reduce it to a free boundary value problem for a quasi-linear elliptic equation in an unbounded domain. The crucial step in our analysis is to derive some uniform a priori estimates in such a domain. Then we apply the fixed point theorem to establish the existence of solutions to the full potential equation. / In chapter 2, we study the short time existence of discontinuous shock front solutions of the pressure gradient system which is the Euler system without inertial terms, where the initial data can have shock discontinuities of arbitrary strength which lie on a given smooth initial surface with arbitrary geometry. These shock solutions are constructed via a classical iteration scheme. The key step is to obtain the uniform stability for the related linearized equation by calculating the Lopatinski's determinant, which enables us to modify the technique of Majda and establish the local existence of solutions to the pressure gradient system without the structural constraints as for the full Euler system. / In this thesis we study two kinds of multi-dimensional shock phenomena for the compressible fluid dynamics. / Xie Feng. / "December 2005." / Adviser: Zhou Ping Xin. / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6446. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 71-80). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

No description available.

In this paper we prove that div(|x|β∇u)+|x|αf(u)=0, inB u = 0 on ∂B has infinitely many solutions when f is superlinear and grows subcritically for u ≥ 0 and up to critically for u less than 0 with 10, 13 N+β−2 N+β−2 We make extensive use of Pohozaev identities and phase plane and energy arguments.

The boundary layer problems in periodic homogenization arise naturally from the quantitative analysis of convergence rates. Formally they are second-order linear elliptic systems with periodically oscillating coefficient matrix, subject to periodically oscillating Dirichelt or Neumann boundary data. In this dissertation, for either Dirichlet problem or Neumann problem, we establish the homogenization results and obtain the nearly sharp convergence rates, provided the domain is strictly convex. Also, we show that the homogenized boundary data is in W1,p for any p ∈ (1,∞), which implies the Cα-Hölder continuity for any α ∈ (0,1).

We deal with a free boundary problem for a nonlinear parabolic equation, which includes a parameter in the free boundary condition. This type of system has been used in models of ecological systems, in chemical reactor theory and other kinds of propagation phenomena involving reactions and diffusion. The main purpose of this dissertation is to show the global existence, uniqueness of solutions and that a Hopf bifurcation occurs at a critical value of the parameter r. The existence and uniqueness of the solution for this problem are shown by finding an equivalent regular free boundary problem to which existence results can be applied. We then show that as the bifurcation parameter r decreases and passes through a critical value rc, the stationary solution loses stability and a stable periodic solution appears. Several figures have been included, which illustrate this transistion. The pascal source program used in the numerical simulation is included in an appendix.

No description available.

Two major areas were studied in this research to achieve more efficient and optimised method for the prediction of ship motion, and this research has two aims. The first aim was to improve an algorithm of the oscillatory problems for strip theory by means of reducing numerical integration using the desingularised method. A new way of distributing point sources was developed by the author in order to solve the boundary problem on the source distribution. Results showed that desingularsation can be utilised on rounded hull shapes. Although the desingularsation process reduces the computational time, the conventional method is more robust and stable due to the simple source panel distribution. The second aim was an investigation of viscous roll damping of ship motion with the influence of forward velocity, and several numerical simulations were developed in order to support wind-tunnel experimentation. The wind tunnel experimentation was conducted by using a 1.2 m NACA6521 modified cylindrical-bulb model to investigate the viscous effect on the rolling motion of the ship. Since viscous damping was very small under restrictions from the experimental condition, a normal method of collecting data of roll motion, in which a device is physically attached on the bulb model, was not suitable. As a solution, remote sensing was utilised to capture the motion picture by a digital video camera. A visual analysis was then conducted to obtain data of the roll motion of the bulb model inside the wind-tunnel test section. Two different numerical simulations were developed under the hypothesis that the forward velocity influences the boundary layer generation to cause viscous roll damping on the ship model hull. The first numerical simulation uses the energy method to produce damping coefficients, and the second numerical simulation requires solving the motion of equation numerically. It was discovered that the increase of forward velocity results in a linear increase of the viscous damping coefficient. The numerical simulation and experimental data agree closely. Therefore, the theory used to predict the viscous roll damping was shown to be reasonably accurate.