Thesis (M.Sc.(Mathematics and Applied Mathematics)) -- University of Pretoria, 2008. / Includes bibliographical references. Available on the Internet via the World Wide Web.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1982. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 53-54).

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 90-93).

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Ibuprofen is a non-steroidal anti-inflammatory drug with three major types of effect: anti-inflammatory, analgesic and antipyretic. Ibuprofen may be administered in a number of different forms via the oral as well as the topical route. Published evidence suggests that topical, unlike oral, non-steroidal anti-inflammatory drugs are associated with few systemic side effects as plasma concentrations are low compared to oral therapy. In some countries it is particularly difficult to obtain human skin for in vitro experimentation and it is therefore important to have alternate biological or synthetic membranes which mimic human skin for diffusion experiments. Synthetic membranes serve as predictive models for topical drug release and in South Africa, shed snake skin is easily obtainable from the many snake parks present in the country. The FDA guidelines were considered when choosing the apparatus to be used in the comparative diffusion study on proprietary ibuprofen-containing topical preparations from three countries and the verification of the usefulness, or otherwise, of shed snake skin as a biological membrane for the assessment of the permeation of ibuprofen. Two diffusion techniques were considered appropriate for the measurement of the amount of ibuprofen released from a topical formulation during in vitro testing. One was the Franz diffusion cell, as modified by Keshary and Chien (88,169) and the other was the European Pharmacopoeia diffusion cell (187). High performance liquid chromatography was used as the analytical technique for the analysis of ibuprofen in aqueous solution using ultraviolet detection at 222 nm. The validated method was applied to the determination of the diffusion of ibuprofen from topical ibuprofen-containing formulations (gels, creams and mousse) through synthetic silicone membrane and shed snake skin biological membrane from four different species. In a study of fifteen topical ibuprofen-containing formulations (gels, creams and mousse) from three countries (South Africa, United Kingdom and France) it was found that there was a trend of products from two countries consistently exhibiting superior diffusion characteristics as well as products from the same two countries consistently exhibiting the lowest diffusion of ibuprofen. Interpretation of the results of these studies demonstrated the importance of employing a combination of statistical analyses and peak integration values when drawing conclusions regarding comparative diffusion characteristics. Shed snake skin has been described as a 'model' membrane, i.e. a membrane which shows similar permeability to human stratum corneum. The results reported here show clearly that, for ibuprofen, the four species of snake produce shed skin with completely different diffusion characteristics when all other conditions are identical. It may well be that there is one particular species of snake which produces shed skin of identical permeability to human stratum corneum, but to describe shed snake skin in general as a model membrane seems incorrect. It is therefore important that if shed snake skin is used as a membrane, the species, skin site and orientation should be reported. The European Pharmacopoeia diffusion apparatus was judged to be the better of the two diffusion techniques assessed for the measurement of the amount of ibuprofen released from a topical formulation during in vitro testing using silicone membranes and for the measurement of the amount of ibuprofen diffusing across the ventral outside orientation of shed skin during in vitro testing, whereas the Franz diffusion apparatus was judged to be better for the measurement of the amount of ibuprofen diffusing across the dorsal outside orientation of shed skin during in vitro testing. However, the choice of this diffusion apparatus must be weighed against the relatively poor reproducibility as compared with the European Pharmacopoeia diffusion apparatus.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1991. / Includes bibliographical references (p. 120-125).

by Cheung Kei Wai. / Parallel title in Chinese. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 119-123). / Acknowledgement --- p.ii / Abstract --- p.iii / List of Abbreviations --- p.vii / List of Figure and Table Captions --- p.viii / List of Publication --- p.xiii / Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- Normal Diffusion --- p.4 / Chapter 1.2 --- Statistical Origin of Anomalous Diffusion --- p.10 / Chapter 1.3 --- Various Models of Diffusion --- p.15 / Chapter 2. --- Scaling Theory of Hopping Transport in One Dimension --- p.21 / Chapter 2.1 --- "Scaling Exponents, the Models and their Solutions" --- p.22 / Chapter 2.2 --- Numerical Simulations --- p.32 / Chapter 2.3 --- Relations to Diffusion on Fractals and Hierarchical Structures --- p.37 / Chapter 2.4 --- Non-Markovian Nature of Results and Quasi-localization Effect --- p.42 / Chapter 3. --- Renormalization Group (RG) Analysis of the Problem --- p.43 / Chapter 3.1 --- The Length Scale Renormalization (LSR) --- p.45 / Chapter 3.2 --- Application of the LSR to Random Barrier Model --- p.52 / Chapter 3.2.1 --- The distribution of the renormalized coupling constants / Chapter 3.2.2 --- Behaviour of W and V under the LSR transformation / Chapter 3.3.3 --- The scaling exponents from LSR / Chapter 3.3 --- Scaling Form of Diffusion Front --- p.66 / Chapter 4. --- Diffusion in Hierarchical Systems --- p.72 / Chapter 4.1 --- Scaling and Probability Densities --- p.75 / Chapter 4.2 --- Exact Renormalization and Lattice Green Function --- p.82 / Chapter 4.3 --- The Range of Diffusion --- p.89 / Chapter 5. --- Biased Diffusion --- p.95 / Chapter 5.1 --- Scaling Theory --- p.98 / Chapter 5.1.1 --- Linear Response / Chapter 5.1.2 --- Diffusion-Drift Crossover / Chapter 5.2 --- Crossover Behaviour under an Additional Bias --- p.103 / Chapter 6. --- Physical Realizations and Related Problems --- p.108 / Chapter 6.1 --- Physical Realizations --- p.109 / Chapter 6.2 --- Equivalence with Random Resistor Network (RRN) Problems --- p.113 / Chapter 7. --- Discussion and Conclusion --- p.116 / References --- p.119 / Chapter Appendix A --- Derivation of Eq. (1.3.8) --- p.124 / Chapter Appendix B --- Derivation of Eq. (1.3.10) --- p.126 / Chapter Appendix C --- Sampling from a Distribution --- p.129 / Chapter Appendix D --- Scaling Form of P (s) in Ordered System --- p.130 / Chapter Appendix E --- Explicit Form of the Lattice Green Function --- p.132

The modelling of transport processes in biological systems is one of the main theoretical challenges in physics, chemistry and biology. This is motivated by their essential role in the emergence of diseases, like tumour metastases, which originate from the spontaneous migration of cancer cells. Thus, improvements in their understanding could potentially pave the way for an outstanding innovation of present-day techniques in medicine. These processes often exhibit anomalous properties, which are qualitatively described by the power-law scaling of their mean square displacement, compared to the linear one of normal diffusion. Such behaviour has been often successfully explained by the celebrated continuous-time random walk model. However, recent experimental studies revealed the existence of both more complicated mean square displacement behaviour and anomalous features in other characteristic observables, e.g. the position-velocity statistics or the two point correlation functions of either the velocity or the position. Thus, in order to understand the anomalous diffusion recorded in these experiments and assess the microscopic processes underlying the observed macroscopic dynamics, one needs to have a complete tool-kit of techniques and models that can be readily compared with the experimental datasets. In this Thesis, we contribute to the construction of such a complete framework by fully characterising anomalous processes, which are described by means of a continuoustime random walk with general waiting time distributions and/or external forces that are exerted both during the jumps (as in the original model) and the waiting times. In the first case we derive both the joint statistics of these processes and their observables, specifically by obtaining a generalised fractional Feynman-Kac formula, and their multipoint correlation functions and employ them to fit the mean square displacement data of diffusing mitochondria. This result supports the experimental relevance of our formalism, which comprises general formulas for several quantities that can provide readily predictable tests to be checked in experiments. In the second case, we characterise the new anomalous processes by means of Langevin equations driven by a novel type of non Gaussian noise, which reproduces the typical fluctuations of a free diffusive continuous-time random walk. For a constant external force, we also obtain the fractional evolution equations of their position probability density function and show that, contrarily to continuous-time random walks, they are weak Galilean invariant, i.e., their position distribution in different Galilean frames is obtained by shifting the sample variable according to the relative motion of the frames. Thus, these processes provide a suitable frame-invariant framework, that could be employed to investigate the stochastic thermodynamics of anomalous diffusive processes.

擴散和對流擴散問題在高度異質性介質以及對流佔優擴散問題一直是一個具挑戰性的問題。眾所周知，對這些問題的數值計算需要相當數量的計算機內存和時間。然而，這些問題的解決方案通常包含一個粗糙的組成部分，它通常是我們感興趣的的量，而這個量一般可以用一個小數目的自由度代表。 / 現今有許多不同方法的目標是在計算粗糙的組件而不計算解的全部細節。在這篇論文中，我們將使用多尺度間斷伽遼金(Galerkin)方法來解決這問題。我們的方法是一種內部處罰間斷伽遼金方法。內部處罰間斷伽遼金方法已被證明是有效和準確的方法用來計算偏微分方程的數值解。我們的方法的一個顯著特點是解空間包含兩個部分，其中一個是傳統的多項式空間，另一個是計算粗糙的組件必不可少的包含分格結構的多尺度空間。我們在這篇論文中證明了該方法的穩定性。此外計算結果中表明，該方法能夠準確地捕捉高度異質性介質中的問題以及邊界和內部層對流佔優問題的解。 / Diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem has been a challenge problem for a long time. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. / There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. In this thesis, we will investigate a multisacle discontinuous Galerkin method to solve the problems. This method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Leung, Wing Tat. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 63-66). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Overview --- p.6 / Chapter 1.2 --- Strong Formulation --- p.8 / Chapter 1.3 --- Weak Formulation --- p.8 / Chapter 1.4 --- Finite Element Method --- p.10 / Chapter 2 --- The discontinuous Galerkin (DG) method --- p.12 / Chapter 2.1 --- Introducion --- p.12 / Chapter 2.2 --- Model problem --- p.13 / Chapter 2.3 --- Interior Penalty Method --- p.14 / Chapter 3 --- The Multiscale Method --- p.18 / Chapter 3.1 --- Introducion --- p.18 / Chapter 3.2 --- Multiscale finite element method --- p.19 / Chapter 3.3 --- Homogenization --- p.20 / Chapter 3.4 --- Convergence --- p.23 / Chapter 3.4.1 --- Convergence for H < ε --- p.23 / Chapter 3.4.2 --- Convergence for H > ε --- p.24 / Chapter 3.5 --- Singularly perturbed convection-diffusion problem --- p.25 / Chapter 4 --- The multiscale-enhanced IPDG method --- p.27 / Chapter 4.1 --- Method description --- p.27 / Chapter 4.2 --- Stability --- p.31 / Chapter 4.3 --- Boundedness --- p.32 / Chapter 4.4 --- Convergence for elliptic problem --- p.37 / Chapter 5 --- Numerical Result --- p.46 / Chapter 5.1 --- Accuracy and convergence tests --- p.46 / Chapter 5.2 --- Some other cases with smooth coefficient --- p.49 / Chapter 5.3 --- Performance with internal or boundary layers --- p.52 / Chapter 5.4 --- Time-dependent problems --- p.55 / Chapter 5.5 --- Performance with more general media --- p.59