Travelling waves for a coupled, singular reaction-diffusion system arising from a model of fractional order autocatalysis with decay: I. Permanent form travelling waves.

This paper considers a model, isothermal, autocatalytic reaction scheme with termination, when the reaction orders, m, n, are fractional, i.e. 0 < m, n < 1. Recent numerical studies by McCabe et al (McCabe P M, Leach J A and Needham D J 2002 On an initial-boundary-value problem for a coupled, singular reaction-diffusion system arising from a model of fractional order chemical autocatalysis and decay Q. J Mech. Appl. Math. 55 511-60), have indicated that the system of coupled, singular (i.e. non-Lipschitz) reaction-diffusion equations which model the evolving reaction are capable of supporting permanent form travelling wave (PTW) structures. We investigate this possibility in detail and compare our results with those of an approximating scalar reaction-diffusion equation, studied by McCabe et al (McCabe P M, Leach J A and Needham D J 1998 The evolution of travelling waves in fractional order autocatalysis with decay. 1. Permanent form travelling waves SIAM J. Appl. Math. 59 870-99; 2000 The evolution of travelling waves in fractional order autocatalysis with decay. H. The initial boundary value problem SIAM J. Appl. Math. 60 1707-48) and the numerical results in McCabe P M et al (2003).