Year

2002

Degree Name

Doctor of Philosophy

Department

School of Mathematics and Applied Statistics

Abstract

complex behaviour of physical phenomena can often be elegantly expressed mathematically. Indeed, applied mathematics is replete with diverse fields of study being expressed by a similar class of equations. Among these are tumour growth, cellular transport and pricing financial derivatives. These quite diverse fields have a striking element of commonality. That is, the observable behaviour can be modelled by either a single or a system of diffusion-convection equations. We use a combination of techniques to elucidate the complex physical phenomena observed in pricing financial derivatives and modelling tumour growth and cellular transport. These techniques include classical symmetry analysis, perturbation expansions and numerical modelling.

begin by developing a mathematical model for the early stages of malignant tumour invasion due to random motility, cellular proliferation, proteolysis and haptotaxis. This model is a complex coupled system of nonlinear diffusionconvection equations. We demonstrate, by using the method of classical symmetry reductions, the existence of an exact analytic solution for malignant tumour invasion. We also find that at early times, in the absence of tumour cell diffusion, a compressed tumour layer is evident. Transient protease production-decay dynamics and diffusion, must be present in order for the tumour concentration peak to be smoothed to realistic levels. In addition, we demonstrate that invasion profiles asymptotically evolve to travelling wave solutions and that kink-like profiles, previously thought to be due to contact inhibition and haptotaxis, can equally be explained by cellular diffusion with a decreasing nonlinear diffusivity.

W e further develop a mathematical model for the in-vitro migration of unbound cells in response to a stimulus of chemoattractant. In the laboratory, this migration is measured in a Boyden chamber, a 48-well chemotaxis chamber or some comparable device. The puzzling saturation effect in chemotactic response, observed in in-vitro studies is replicated by our mathematical model and is due to the simple nonlinear coupling between chemotaxis and diffusion, even before receptor saturation or attractant absorption are considered. We find that the decelerating wave of advance, due to chemotaxis alone, fails short of the opposite wall of the chemotaxis chamber, but it causes an overshoot in mass transport before diffusion effects a flux reversal.

mathematical model for pricing financial derivatives involves a diffusionconvection- reaction equation. We are able, by making use of symmetry invariants, to construct new exact analytic solutions for the price of swaps and coupon bearing bonds. These solutions exhibit desirable features such as mean-reversion and power-law dependence on interest rates.

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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.