## Year

2013

## Degree Name

Doctor of Philosophy

## Department

School of Mathematics and Applied Statistics

## Recommended Citation

Alharthi, Muteb Ruddah, Semi-analytical solutions for autocatalytic reaction-diffusion equations, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2013. https://ro.uow.edu.au/theses/3950

## Abstract

In this thesis autocatalytic reactions in a diffusion cell are considered. Cubic autocatalytic reactions have proved a useful test-bed for examining static and dynamic stability of chemical systems. Various extensions and aspects of this model are considered such as mixed quadratic-cubic reactions, the effect of a precursor chemical, a circularly symmetric cell and delay feedback control.

For all of these models, the Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for the reaction-diffusion cell. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space in which Hopf bifurcations can occur is found by a local stability analysis of the semi-analytical model.

In chapter 2 mixed quadratic and cubic reactions are considered. The effect of varying the relative importance of the quadratic and cubic reaction terms and the diffusion coefficient of the two species are examined in detail with much additional complexity, involving bifurcation patterns and stability, found. In chapter 3 the effect of a precursor chemical is examined. The reactant is supplied by two mechanisms, diffusion via the cell boundaries and decay of an abundant precursor chemical present in the reactor. The effect of varying the relative importance of the precursor chemical is examined in detail. In chapter 4 a circularly symmetric reaction diffusion cell or annulus is considered. This allows geometric effects, such as varying the width of the annulus, to be examined. This is one of the first studies to consider reactions in a circularly symmetric annulus. In chapter 5 feedback control with delay is examined by varying the boundary reservoir concentrations in response to the concentrations in the centre of the reactor. The effect of varying the strength of the feedback and the delay are both considered. This study illustrates that feedback control can modify the stability of the system, in terms of stabilizing and destabilizing regions of parameter space. A new technique for finding the regions of parameter space is used in the case of delay, as expressions for the degenerate Hopf points can not be obtained.

For each of the models considered steady-state profiles, bifurcation diagrams, parameter space stability maps and limit-cycle dynamics are drawn. The numerical solutions of the governing partial differential equations are also obtained for comparison, and show the usefulness and accuracy of the semi-analytical results. The methods used here have a general applicability and can be used to develop semi-analytical solutions to other chemical systems for which diffusion is important.

## FoR codes (2008)

0102 APPLIED MATHEMATICS

**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.**