Matched Asymptotic Expansions in Reaction-Diffusion Theory
The idea for this monograph was born out of the desire to collate the results from two distinct strands of the authors' research with the common theme of the application of the method of matched asymptotic expansions to problems arising in reaction-diffusion theory. In Part I, the method ofmatched asymptotic expansions (MAE) is used to obtain the complete structure ofthe solution to reaction-diffusion equations of the Fisher-Kolmogorovtype for large-r (dimensionlesstime), whichexhibit the formation ofa permanent form travelling wave(PTW) structure. In particular, the wavespeed for the large-t PTW, the correction to the wave speed and the rate of convergence of the solution onto the PTW are obtained. The primary focus of Chapters 2-4is the scalar Fisher-Kolmogorov equation with either the generalized Fisher nonlinearity or the mth-order (m > 1) Fisher nonlinearity while in Chapter 5 the analysis is extended by consideration of a system of Fisher-Kolmogorov equations. The methodology developed is flexibleand has wide applicability to scalar and systems of Fisher-Kolmogorov equations in one or higher spatial dimensions. The method of matched asymptotic expan sions has also been used successfully to give information about the structure and propagation speed of accelerating phase wave (PHW) structures which can evolvein reaction-diffusion equations (see Needham and Barnes ) and nonlinear diffusion equations of Fisher-Kolmogorov type. The approach pre sented in this part of the monograph is based on the results obtained in the series of papers by Leach and Needham ,,, Leach, Needham and Kay ,, and Smith, Needham and Leach .
Presents completely new methodologies, based on matched asymptotic expansions Can be used both as a handbook and as a detailed description of the methodologies