The first four papers of the present Part, and all papers in the previous Part, together constitute a "Special Issue" of the
journal devoted to the theme of nonlinear diffusion, using a liberal interpretation of this terminology. The issue contains
contributions from diverse areas, such as flow in porous media, nonlinear diffusion with absorption, microwave heating,
dispersion of pollutants, and others, all of which reveal a rich area of current research.
Professor Ian Sneddon often remarked that when he embarked upon his research career, there existed an endless
array of "nice" linear problems which willingly submitted to the classical formal devices of linear mathematics. One had
only to choose a particular problem and follow a well-trodden path which led to a simple elegant conclusion. In contrast,
when I commenced my research career, there remained only the technically difficult linear problems, and an atmosphere
existed that the solvable linear problems had all been done. In this environment, the nonlinear problems emerged as the
real challenge, and the early books by Professor Bill Ames constituted the only systematic collections of nonlinear
mathematical procedures. At about the same time and in a number of countries, known one-parameter Lie transformation
groups were exploited to determine special solutions of nonlinear partial differential equations. Subsequently there
arose renewed interest in the use of Backlund and related transformations for the solution of nonlinear boundary-value
problems. Such mathematical developments were not always universally applauded, and were often criticised because
frequently the methods lacked the freedom to impose arbitrary initial and boundary data. That is, many nonlinear
analytical techniques imply particular initial and boundary data, and it is usually a fluke if they coincide with an
interesting physical problem. Thus for a time the isolated and ad-hoc techniques of nonlinear mathematical analysis seemed
doomed to languish as curiosities.
However, during this period, numerical techniques and computing capacity developed to such a high degree that many real
nonlinear boundary-value problems could be solved readily on the computer, and to a certain extent these developments
removed the necessity of concentrating on and solving real physical problems. Today the interest and development in special
devices in nonlinear mathematics proceeds unabated. This is because one of the key features of nonlinearity is the range and
variety of physical response, which is often bizarre and unexpected, but which is frequently embodied in the simplest of
exact solutions. Nonlinear diffusion is characterised by phenomena such as "blow-up", "extinction" and "waiting time"
behaviour, all of which can be readily illustrated by simple exact results. Even though the theoretical development is often
running well ahead of experimental confirmation, we have, in all probability, only scratched the surface of the subject. No
doubt there remain interesting physical phenomena, embodied in our models, which have yet to be identified from either a
theoretical or practical perspective. When you consider the fundamental role enjoyed by the linear heat-diffusion equation
in science and engineering, consider how much more variety of response is embodied in nonlinear diffusion models which
constitute a far closer approximation to reality. The subject of nonlinear diffusion will be with us, at least for a
while, and I trust this special issue serves to generate new interest in this important area of applied mathematics.
I am grateful to the editor of the journal for originally suggesting the special issue and also to the authors and
referees for their contributions to its production.
J. M. Hill, Guest Editor
Editor JAMSB(E): editor at anziamj.austms.org.au
WWW Administrator: webmaster at anziamj.austms.org.au