Yim Man Yi = 亞擴散雙穩分數福克-普朗克系統的線性響應及隨機共振和有色噪音對雙穩系統所引起的效應 / 嚴敏儀. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 85-89). / Text in English; abstracts in English and Chinese. / Yim Man Yi = Ya kuo san shuang wen fen shu Fuke-Pulangke xi tong de xian xing xiang ying ji sui ji gong zhen he you se zao yin dui shuang wen xi tong suo yin qi de xiao ying / Yan Minyi. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Brownian motion and anomalous dynamics --- p.1 / Chapter 1.2 --- White and colored noises --- p.3 / Chapter 2 --- Linear response theory of sub diffusive Fokker-Planck systems --- p.6 / Chapter 2.1 --- Introduction to subdiffusive Fokker-Planck systems --- p.6 / Chapter 2.2 --- Spectral density --- p.8 / Chapter 2.3 --- Linear response --- p.11 / Chapter 2.4 --- Signal-to-noise ratio (SNR) --- p.13 / Chapter 2.5 --- Stochastic energy --- p.14 / Chapter 3 --- Perturbation due to a sinusoidal signal --- p.16 / Chapter 3.1 --- Presence of an external sinusoidal forcing --- p.16 / Chapter 3.1.1 --- PDF and linear reponse --- p.16 / Chapter 3.1.2 --- Phase lag --- p.19 / Chapter 3.1.3 --- SNR --- p.19 / Chapter 3.1.4 --- Stochastic energy --- p.23 / Chapter 3.2 --- Presence of a sinusoidally time-varying diffusion coefficient --- p.23 / Chapter 4 --- Perturbation due to a rectangular signal --- p.26 / Chapter 4.1 --- Linear response to a rectangular pulse --- p.26 / Chapter 4.2 --- Perturbation due to a periodic rectangular signal --- p.28 / Chapter 4.2.1 --- Linear response --- p.29 / Chapter 4.2.2 --- SNR --- p.31 / Chapter 4.2.3 --- Stochastic energy --- p.33 / Chapter 4.3 --- Comparison with the response to a sinusoidal driving force --- p.35 / Chapter 5 --- The Effects of Colored Noise on the Stationary Probability Distribution --- p.38 / Chapter 5.1 --- Formulation of a general colored noises-driven system --- p.39 / Chapter 5.2 --- Approximation schemes for a colored noise --- p.42 / Chapter 5.2.1 --- Decoupling approximation --- p.42 / Chapter 5.2.2 --- UCNA --- p.45 / Chapter 5.2.3 --- Small r approximation . --- p.46 / Chapter 5.2.4 --- Presence of an additive noise: g(x) = 1 --- p.47 / Chapter 5.2.5 --- Presence of a multiplicative noise: g(x) = x --- p.52 / Chapter 5.3 --- Approximation scheme for two colored noises --- p.53 / Chapter 5.3.1 --- Presence of two additive noises --- p.55 / Chapter 5.3.2 --- Presence of an additive noise and a multiplicative noise --- p.55 / Chapter 5.4 --- Unimodal-bimodal transitions --- p.61 / Chapter 6 --- A stochastic genetic regulatory transcription model with colored noises --- p.68 / Chapter 6.1 --- Biological background --- p.69 / Chapter 6.2 --- Effect of a single noise --- p.72 / Chapter 6.3 --- Effects of two noises --- p.75 / Chapter 6.4 --- Biological implications of colored noises in the model --- p.81 / Chapter 7 --- Conclusions --- p.82 / Bibliography --- p.85 / Chapter A --- Mittag-Leffler function --- p.90 / Chapter B --- H-function (Fox function) --- p.92 / Chapter C --- Operator algebra --- p.94 / Chapter D --- Mean first passage time --- p.97

Ces travaux concernent deux équations paraboliques, dégénérées et non-locales. La première équation est une équation de films minces fractionnaire et la deuxième est une équation des milieux poreux fractionnaire. La présentation des problèmes, les résultats existants dans la littérature, ainsi que le résumé de nos résultats font l'objet de l'introduction. Le deuxième chapitre est consacré à la présentation de la méthode de De Giorgi utilisée pour montrer la régularité Hölder des solutions des équations elliptiques. On présente de plus les résultats utilisant cette approche dans les cas paraboliques local et non-local. Dans le troisième chapitre, on montre l'existence de solutions faibles d'une équation des films minces fractionnaire. C'est une équation parabolique, dégénérée, non-locale d'ordre $alpha+2$ où $0 < alpha < 2$. C'est une généralisation d'une équation étudiée par Imbert et Mellet en 2011 pour $alpha = 1$. Pour construire les solutions, on passe par un problème régularisé. En utilisant les injections de Sobolev, on passe à la limite pour trouver des solutions faibles. Vu la différence des injections de Sobolev, on distingue deux cas $0 <alpha < 1$ et $1 leq alpha < 2$. Dans les deux cas on démontre que la solution est positive si la condition initiale l'est. Le quatrième chapitre concerne une équation des milieux poreux fractionnaire. On montre la régularité Hölder de solutions faibles positives satisfaisant des estimées d'énergie. D'abord, on montre l'existence de solutions faibles qui satisfont des estimées d'énergie. On distingue deux cas $0 <alpha < 1$ et $1 leq alpha < 2$ à cause de problème de divergence. Puis on démontre les lemmes de De Giorgi qui sont des lemmes de réduction de l'oscillation d'en dessus et d'au-dessous. Ces deux lemmes ne suffisent pas pour montrer la régularité Hölder. On a besoin d'améliorer le résultat du lemme de réduction de l'oscillation d'en dessus. Donc, on passe par un lemme des valeurs intermédiaires et on montrer un lemme de réduction de l'oscillation d'en dessus amélioré. Enfin, on montre la régularité Hölder des solutions en utilisant la propriété scaling de ces solutions / In this thesis, we study two degenerate, non-local parabolic equations, a fractional thin film equation and a fractional porous medium equation. The introduction contains a presentation of problems, the previous results in the literature and a brief presentation of our results. In the second chapter, we present a short overview of the De Giorgi method used to prove Hölder regularity of solutions of elliptic equations. Moreover, we present the results using this approach in the local and non-local parabolic cases. In the third chapter we prove existence of weak solutions of a fractional thin film equation. It is a non-local degenerate parabolic equation of order $alpha + 2$ where $0 < alpha < 2$. It is a generalization of an equation studied by Imbert and Mellet in 2011 for $alpha = 1$. To construct these solutions, we consider a regularized problem then we pass to the limit using Sobolev embedding theorem, that's why we distinguish two cases $0 < alpha < 1$ and $1 leq alpha < 2$. We also prove that the solution is positive if the initial condition is so. The fourth chapter is dedicated for a fractional porous medium equation. We prove Hölder regularity of positive weak solutions satisfying energy estimates. First, we prove the existence of weak solutions that satisfy energy estimates. We distiguish two cases $0 < alpha < 1$ and $1 leq alpha < 2$ because of divergence problems. The we prove De Giorgi Lemmas about oscillation reduction from above and from below. This is not suffisant. We need to improve the lemma about oscillation reduction from above. So we pass by an intermediate values lemma and we prove an improved oscillation reduction lemma from above. Finally, we prove Hölder regularity of solutions using the scaling property

Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The differential-operator approximation to the linear Boltzmann operator (the Master equation operator) has been studied by several authors. In 1960 Siegel proposed a systematic approach called the CD expansion. He represents the approximating series in terms of creation and destruction operators for Hermite functions. In this thesis we study the physical meaning of a small parameter which usually exists in the CD expansion and which ensures the convergence of the series. We also establish the CD-expansion formalism for N-dimensional processes and initiate the study of the CD-expansion of the linear Boltzmann collision operator in the kinetic theory of gases. In the case of one-dimensional processes; the models we study are the density fluctuations with a particle reservoir of finite volume, Alkemade's diode model, and Rayleigh disk. We find that the expansion parameter is the ratio of the average microscopic agitation interval to the macroscopic relaxation time. We further prove that this ratio is equal to the ratio of the average variance of the discontinuity of the random process determined by the linear. Boltzmann operator to the variance of the macroscopic observable at equilibrium. Since the CD expansion is an expansion with respect to the parameter of discontinuity, the expansion series reduces to the Fokker-Planck operator in the limiting case where the parameter becomes zero. In the N-dimensional formalism, we use tensor Hermite polynomials and find a formalism valid for processes of any finite dimensionality. In extending the study to the kinetic theory of gases, we establish a method of obtaining derivate moments directly from the collision operator, and obtain a formula for the Hermite coefficients of derivate moments for an arbitrary force field. We propose the CD hypothesis: The terms of the CD expansion are homogeneous and of successively increasing order in the parameter of discontinuity of the process. This hypothesis I holds for all the models we study in the one-dimension case. In three-dimensional collision processes it holds for an intermolecular force field obeying an inverse power law and for rigid spheres. Beyond these cases, the necessary and sufficient conditions for the hypothesis are rather complicated. A sufficient condition is, however, that the scattering cylinder (scattering cross section multiplied by the magnitude of the relative velocity) be a homogeneous function of the magnitude of the relative velocity. In addition to the general results mentioned in the above, we obtain a number of particular results. Special cases of our density fluctuation model are the density fluctuations studied by van Kampen (1961), and Ehrenfest's urn model. We also introduce an isotropic Maxwellian particle which corresponds to s-wave scattering in wave mechanics, and which yields the CD expansion in a diagonal form. / 2031-01-01

This thesis explores one kind of equation used to model the physics behind one beam with two ends fixed. Initially, Woinowsky Krieger sets a nonlinear partial differential equation (PDE) model by attaching one nonlinear term to the classic linear beam equation. From Zdzislaw Brezezniak, Bohdan Maslowski, Jan Seidler, they demonstrate this model mixed with one Brownian motion term describing random fluctuation. After stochastic modifications, this model becomes more accurate to the behaviors of beam vibrations in reality, and theoretically, the solution has better properties. In this thesis, the model includes more complex noises which cover the condition of random uncontinuous disturbance in the language of Poisson random measure. The major breakthrough of this work is the proof of existence and uniqueness of solutions to this stochastic beam equation and solves the flaws of previous work on proof.

Lau Ching Yan Ada = 耗散系統中的廣義Sturm-Liouville理論 / 劉正欣. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 156-157). / Text in English; abstracts in English and Chinese. / Lau Ching Yan Ada = Hao san xi tong zhong de guang yi Sturm-Liouville li lun / Liu Zhengxin. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Vibrational motion in physics --- p.1 / Chapter 1.2 --- Normal modes of vibration --- p.2 / Chapter 1.3 --- Boundary conditions --- p.4 / Chapter 1.4 --- The wave equation --- p.6 / Chapter 1.4.1 --- Mechanical waves --- p.7 / Chapter 1.4.2 --- Electromagnetic waves --- p.9 / Chapter 1.5 --- General form of the wave equation --- p.10 / Chapter 1.5.1 --- V(x) as a restoring force --- p.11 / Chapter 1.5.2 --- V(x) in gravitational waves --- p.13 / Chapter 1.5.3 --- V(x) by transformation --- p.16 / Chapter 2 --- Sturm-Liouville systems --- p.18 / Chapter 2.1 --- Introduction --- p.18 / Chapter 2.2 --- Differential operators --- p.19 / Chapter 2.2.1 --- Introduction --- p.19 / Chapter 2.2.2 --- Adjoint operators --- p.20 / Chapter 2.2.3 --- Self-adjoint operators --- p.21 / Chapter 2.2.4 --- More examples --- p.24 / Chapter 2.3 --- Sturm-Liouville boundary-value problems --- p.27 / Chapter 2.4 --- Sturm-Liouville theory --- p.28 / Chapter 2.4.1 --- Real eigenvalues --- p.29 / Chapter 2.4.2 --- Orthogonal eigenfunctions --- p.30 / Chapter 2.4.3 --- Completeness of eigenfunctions --- p.31 / Chapter 2.4.4 --- Interlacing zeros of the eigenfunctions --- p.33 / Chapter 2.5 --- Applications of Sturm-Liouville theory --- p.35 / Chapter 2.5.1 --- Vibrations of a string --- p.36 / Chapter 2.5.2 --- The hydrogen atom --- p.40 / Chapter 3 --- Wave equation with damping --- p.46 / Chapter 3.1 --- Statement of problem --- p.46 / Chapter 3.1.1 --- The equation --- p.46 / Chapter 3.1.2 --- The operator --- p.48 / Chapter 3.1.3 --- Non-self-adjointness --- p.49 / Chapter 3.2 --- Eigenfunctions and Eigenvalues --- p.51 / Chapter 3.3 --- The completeness problem --- p.53 / Chapter 4 --- Green's function solution --- p.55 / Chapter 4.1 --- Introduction --- p.55 / Chapter 4.2 --- Green's function solution --- p.56 / Chapter 4.3 --- Fourier transform --- p.58 / Chapter 4.4 --- Inverse Fourier transform --- p.61 / Chapter 5 --- Proof of completeness --- p.66 / Chapter 5.1 --- WKB approximation --- p.66 / Chapter 5.2 --- "An upper bound for \G(x,y,w)e~iwt\ " --- p.68 / Chapter 5.3 --- Proof of completeness --- p.72 / Chapter 5.3.1 --- The limit when R→∞ --- p.72 / Chapter 5.3.2 --- Eigenfunction expansion --- p.76 / Chapter 6 --- The bilinear map --- p.80 / Chapter 6.1 --- Introduction --- p.80 / Chapter 6.2 --- Evaluation of J1(wj) --- p.82 / Chapter 6.3 --- Self-adjointness of H --- p.84 / Chapter 6.4 --- Properties of the map --- p.87 / Chapter 7 --- Applications --- p.89 / Chapter 7.1 --- Eigenfunction expansion --- p.89 / Chapter 7.2 --- Perturbation theory --- p.94 / Chapter 7.2.1 --- First and second-order corrections --- p.95 / Chapter 7.2.2 --- Example --- p.97 / Chapter 7.2.3 --- Example (Constant r) --- p.102 / Chapter 8 --- Critical points --- p.104 / Chapter 8.1 --- Introduction --- p.104 / Chapter 8.2 --- Conservative cases (Γ = 0) --- p.105 / Chapter 8.3 --- Non-conservative cases (Constant r) --- p.107 / Chapter 8.4 --- Critical points away from imaginary axis --- p.108 / Chapter 9 --- Jordan block and applications --- p.114 / Chapter 9.1 --- Jordan basis --- p.114 / Chapter 9.2 --- An analytical example --- p.117 / Chapter 9.2.1 --- Solving for the extra basis function --- p.117 / Chapter 9.2.2 --- Freedom of choice --- p.118 / Chapter 9.2.3 --- Interpolating function --- p.120 / Chapter 9.3 --- A numerical example --- p.122 / Chapter 9.3.1 --- "Solving for f2,1 " --- p.124 / Chapter 9.3.2 --- Interpolating function --- p.126 / Chapter 9.4 --- Jordan basis expansion --- p.127 / Chapter 9.5 --- Perturbation theory near critical points --- p.131 / Appendices --- p.142 / Chapter A --- WKB approximation --- p.142 / Chapter B --- Green's function (Discontinuous V(x)) --- p.145 / Chapter B.l --- Finite discontinuouity in V(x) --- p.145 / Chapter B.1.1 --- Green's function --- p.145 / Chapter B.1.2 --- "Behaviour of the extra phases Φ, Φ " --- p.147 / Chapter B.2 --- Delta function in --- p.148 / Chapter B.2.1 --- Green's function --- p.148 / Chapter B.2.2 --- "Behaviour of the extra phases Φ, Φ " --- p.150 / Chapter C --- Dual basis --- p.151 / Chapter C.1 --- Matrix representation --- p.152 / Chapter C.2 --- Relation with bilinear map --- p.153 / Chapter C.3 --- Construction of dual basis --- p.154 / Bibliography --- p.156

Chow Cheuk Wang = 含粒子阱之福克-普朗克方程及亞擴散分數福克-普朗克方程之研究 / 周卓宏. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 70-72). / Text in English; abstracts in English and Chinese. / Chow Cheuk Wang = Han li zi jing zhi Fuke-Pulangke fang cheng ji ya kuo san fen shu Fuke-Pulangke fang cheng zhi yan jiu / Zhou Zhuohong. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Derivation of the Fokker-Planck equation --- p.5 / Chapter 2.1 --- Diffusion equation --- p.5 / Chapter 2.2 --- Kramers-Moyal Equation --- p.7 / Chapter 2.3 --- Fokker-Planck Equation --- p.9 / Chapter 2.3.1 --- Eigenfunction Expansion --- p.10 / Chapter 2.3.2 --- Mapping FPE to a pseudo-Schrodinger equation --- p.11 / Chapter 3 --- Conventional Fokker-Planck equation with sinks --- p.15 / Chapter 3.1 --- Propagator with sinks --- p.16 / Chapter 3.1.1 --- One-sink propagator --- p.17 / Chapter 3.1.2 --- Two-sink propagator --- p.18 / Chapter 3.2 --- Survival probability --- p.19 / Chapter 3.3 --- Expectation value of the position --- p.22 / Chapter 3.4 --- Mean survival time --- p.25 / Chapter 4 --- Fractional Fokker-Planck equation with sinks --- p.38 / Chapter 4.1 --- Fractional diffusion equation --- p.39 / Chapter 4.2 --- Propagator of the subdiffusive system --- p.41 / Chapter 4.3 --- Survival probability and expectation value of the position --- p.43 / Chapter 4.4 --- Mean survival-time distribution --- p.47 / Chapter 5 --- Boundary value problems for diffusion and subdiffusion --- p.53 / Chapter 5.1 --- Diffusion in a linear potential U(x) = Fx --- p.54 / Chapter 5.2 --- Two absorbing boundaries --- p.55 / Chapter 5.3 --- One absorbing boundary and one reflecting boundary --- p.58 / Chapter 5.4 --- Two reflecting boundaries --- p.59 / Chapter 6 --- Summary --- p.68 / Bibliography --- p.70 / Chapter A --- Laplace transform and the method of Abate and Whitt --- p.73 / Chapter B --- Mittag-Leffler function and its two-point Pade approximant --- p.77

Bai Yun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 34-36). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- The Analysis of a Two-Level SEM with Group Specific Variables in LISREL --- p.4 / Chapter 2.1 --- The Model --- p.4 / Chapter 2.2 --- An Augmented Model --- p.5 / Chapter 2.3 --- Implementation in LISREL --- p.7 / Chapter 2.4 --- Simulation --- p.9 / Chapter 2.4.1 --- The Simulation Design --- p.9 / Chapter 2.4.2 --- Methods of Evaluation --- p.10 / Chapter 2.4.3 --- Simulation Results --- p.12 / Chapter 2.5 --- A Comparison to Mplus --- p.14 / Chapter 2.6 --- Empirical Demonstration: Multi-source Performance Appraisals --- p.14 / Chapter 3 --- Implementing Two level SEM with Cross-level Covariance Structures in Mx --- p.16 / Chapter 3.1 --- Two level Model Specifications with a Cross-level Covariance Structure --- p.17 / Chapter 3.2 --- An Illustrative Example --- p.20 / Chapter 3.3 --- Mx Simulation Design --- p.22 / Chapter 3.4 --- Simulation Results --- p.23 / Chapter 3.4.1 --- Accuracy of Parameter Estimates --- p.23 / Chapter 3.4.2 --- Accuracy of Standard Error Estimates --- p.24 / Chapter 3.4.3 --- Distribution of Goodness-of-fit Statistics --- p.24 / Chapter 3.5 --- Enlarged Mx Model --- p.24 / Chapter 3.5.1 --- Mx Model with Enlarged Xgi --- p.25 / Chapter 3.5.2 --- Mx Model with Enlarged Ng --- p.26 / Chapter 4 --- LISREL Sampling --- p.27 / Chapter 4.1 --- LISREL Sampling Simulation Design --- p.27 / Chapter 4.2 --- Simulation Results --- p.28 / Chapter 4.2.1 --- Accuracy of Parameter Estimates --- p.29 / Chapter 4.2.2 --- Accuracy of Standard Error Estimates --- p.30 / Chapter 4.2.3 --- Distribution of Goodness-of-fit Statistics --- p.30 / Chapter 5 --- Discussion --- p.31 / Appendices --- p.37 / Appendix 1 LISREL Sample Program --- p.37 / Appendix 2 LISREL Syntax for an ALL-Y Model --- p.38 / Appendix 3 LISREL Data Set Up --- p.39 / Appendix 4 Mx Sample Program --- p.40 / List of Figures / Chapter 1 --- The Augmented Two-level Model --- p.41 / Chapter 2 --- Results of the Performance Appraisal Example --- p.42 / Chapter 3 --- Two-level Model with a Cross-level Structure --- p.43 / Chapter 4 --- QQ-plot for P1-P6 --- p.44 / Chapter 5 --- QQ-plot for M1-M6 --- p.45 / List of Tables / Chapter 1 --- Simulation Conditions Associated with Each Pattern --- p.46 / Chapter 2 --- Simulation Results: Accuracy of Parameter Estimates --- p.47 / Chapter 3 --- Simulation Results: Precision of Standard Error Estimates --- p.48 / Chapter 4 --- Simulation Results: The Goodness-of-fit(GOF) Statistics --- p.49 / Chapter 5 --- Analysis of the Performance Appraisal Example --- p.49 / Chapter 6 --- Simulation Results: Mplus vs. LISREL-Parameter Estimates(l) --- p.50 / Chapter 7 --- Simulation Results: Mplus vs. LISREL-Parameter Estimates(2) --- p.51 / Chapter 8 --- Simulation Results: Mplus vs. LISREL-SE Estimates (Ratio) --- p.52 / Chapter 9 --- Simulation Results: Mplus vs. LISREL-GOF Statistics --- p.53 / Chapter 10 --- Mx Illustrative Example Results --- p.53 / Chapter 11 --- Mx Simulation Patterns --- p.53 / Chapter 12 --- Mx Simulation Results: Accuracy of Parameter Estimates --- p.54 / Chapter 13 --- Mx Simulation Results: MARB for Parameter and S.E. Estimates --- p.54 / Chapter 14 --- Mx Simulation Results: Goodness-of-fit Statistics --- p.55 / Chapter 15 --- Mx Simulation Results for M5 --- p.55 / Chapter 16 --- Mx Simulation Results for M5 and M6: Goodness-of-fit Statistics --- p.56 / Chapter 17 --- Mx Simulation Results for M6 --- p.56 / Chapter 18 --- LISREL Sampling: Simulation Patterns --- p.56 / Chapter 19 --- LISREL Sampling: Simulation Results for LI to L3 --- p.57 / Chapter 20 --- LISREL Sampling: Simulation Results for L4 to L6 --- p.58 / Chapter 21 --- LISREL Sampling: MARB for Parameter and S.E. Estimates --- p.59 / Chapter 22 --- LISREL Sampling: Goodness-of-fit Statistics --- p.59

Made available in DSpace on 2014-06-11T19:22:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-02-27Bitstream added on 2014-06-13T20:28:32Z : No. of bitstreams: 1 santos_gr_me_sjrp.pdf: 399304 bytes, checksum: 6932f6a8d97705e593070d265e935262 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Este trabalho tem como objetivo aplicar o método das aproximações sucessivas na demonstração do Teorema do ponto fixo de Banach e em resultados que garantem a existência e unicidade de soluções de equações diferenciais definidas em espaços de Banach / This work aims to apply the method of successive approximations in the proof of Banach fixed point Theorem and in results that guarantee the existence and uniqueness of solutions of differential equations defined in Banach spaces

一維的無粘Burgers方程是最簡單的非線性雙曲守恆型方程，在本篇論文中，我們提出一個交錯間斷伽遼金方法去解Burgers方程。交錯間斷伽遼金方法融合了標準有限元方法和標準間斷伽遼金方法，此方法會求兩個間斷函數的解，而這對函數間斷的地方是不同的，所以在其中一個函數間斷的位置，另外的函數加強了該函數的連續性。對於Burgers方程來說，要求的解及通量組成了一對交錯對，我們將構造這個交錯間斷伽遼金格式和證明這格式是能量守恆的。 / 典型Burgers方程的解常存有衝擊波和間斷的地方，在這些情況下，我們的格式不再是能量守恆，並且出現了數值振蕩的問題，我們會提出兩個方案去除掉數值解中的數值振蕩。第一個方法是把一個人工的擴散性通量加在數值格式裏，這個人工的擴散性通量是從一個解粘性Burgers方程的交錯間斷伽遼金格式中求得的，這個格式的構造過程跟構造原格式的過程是類似的。為確保數值解的準確度，擴散性通量只會在存有數值振蕩的地方才加上。第二個方法是一個全變差正則化方法，在某些保留數值解的準確性的條件下，振蕩性數值解的全變差會被減至最小。這個步驟只用於存在振蕩的地方，以減小計算成本和多餘的誤差。另外，處理最小化問題時會用到Bregman算法。本篇論文將記述有關這兩個方法的細節和數值驗証。 / The 1D inviscid Burgers' equation is the simplest nonlinear hyperbolic conservation law. In this thesis, a staggered discontinuous Galerkin method for the Burgers' equation is proposed. Staggered discontinuous Galerkin method is a kind of DG method that compromise conforming finite element method and standard DG method. Two unknown functions that are discontinuous at different points are solved, thus extra continuity is imposed at the points of discontinuity of the discontinuous function by the staggered counter part. For the Burgers' equation, the unknown function and the flux form the staggered pair. We will derive this staggered DG scheme and show that the scheme is energy conserving. / Typical problems concerning the Burgers' equation involve shock waves and discontinuous solutions. In such cases, the scheme is no longer energy conserving and the problem of numerical oscillations arises. Two approaches are presented to eliminate the numerical oscillations in the solution. The rst one is based on adding an artificial diffusive flux to the scheme. The artificial diffusive flux is derived from a staggered DG scheme for the viscid Burgers' equation for which the derivation is similar. To preserve accuracy, the artificial diffusive flux is added only at regions with oscillations. The second approach is a TV regularization method. The total variation of the oscillatory numerical solution is minimized under certain constraints that preserve the accuracy of the solution. To reduce computation cost and redundant error, the TV minimization process is induced locally in regions with oscillations. Bregman algorithm is applied for numerical implementation of the minimization problem. Detailed description of the two methods and the numerical results are presented in this thesis. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Chan, Hiu Ning. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 71-73). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.7 / Chapter 2 --- Inviscid scheme --- p.10 / Chapter 2.1 --- Space discretization and element spaces --- p.10 / Chapter 2.2 --- Derivationofinviscidscheme --- p.11 / Chapter 2.3 --- Conservationofenergy --- p.13 / Chapter 2.4 --- Piecewiseconstantcase --- p.17 / Chapter 2.5 --- Problemwithdiscontinuity --- p.18 / Chapter 3 --- Mixed method --- p.21 / Chapter 3.1 --- Viscidscheme --- p.22 / Chapter 3.1.1 --- Derivation of viscid scheme --- p.22 / Chapter 3.1.2 --- Conservationofenergy --- p.24 / Chapter 3.1.3 --- Piecewiseconstantcase --- p.26 / Chapter 3.2 --- Relations between the inviscid scheme and the viscid scheme --- p.27 / Chapter 3.3 --- Mixed method with piecewise constant elements --- p.32 / Chapter 3.4 --- Mixed method with piecewise linear elements --- p.35 / Chapter 3.5 --- Numericalresults --- p.40 / Chapter 3.5.1 --- Figures --- p.40 / Chapter 3.5.2 --- Error --- p.49 / Chapter 4 --- A local TV regularization method --- p.56 / Chapter 4.1 --- LocalTVminimizationproblem --- p.56 / Chapter 4.2 --- Oscillationvector --- p.57 / Chapter 4.3 --- Methoddescription --- p.59 / Chapter 4.4 --- Implementation --- p.61 / Chapter 4.5 --- Remarkon’global’method --- p.63 / Chapter 4.6 --- Numericalresults --- p.63 / Chapter 5 --- Conclusion --- p.69 / Bibliography --- p.71

Lau Shek Kwan Mark. / "September 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 118-124). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.