Degree Name

Doctor of Mathematics


School of Mathematics and Applied Statistics


This thesis investigates diffusion equations in polar geometries that do not assume azimuthal symmetry. Many of these investigations can be traced back to pioneers like Carswell and Jaeger, but in almost every case the cylindrical or spherical models assume θ independence. Until the work of the Losic Group at University of South Australia was discovered, it was unclear what practical value an investigation of non-axisymmetric mixed boundary problems would have. The cylindrical diatom structure, with its windows open to diffusion, gave the first clear indication that a promising, emergent technology may offer some application.

The introductory chapter attempts to summarize some of the broad array of publications on the subject in the past few years. The burgeoning list of these titles is testimony to the traction that the field of controlled release is currently experiencing. Several important diversions are entered into. These include the importance of the parameter for diffusivity, D, and its development through chemistry and materials science. It also examines the effects of the initial and boundary conditions that are so instrumental to the applicability of the mathematics to this particular field. It concludes with a review of current research into the use of diatom microfrustules as delivery devices for controlled release applications.

The subsequent chapters form a review of one dimensional polar-based initial and boundary value problems. While these are available in many diverse textbooks, none of them contained all the results included herein. In each of the chapters, the methods of variable separation, Laplace transform and Green’s function solutions are examined. The formulae for the pivotal pharmaceutical quantities of outward surface flux and mass transfer are derived to conclude each chapter.

FoR codes (2008)




Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.