Analytical Solution of Cubic Autocatalytic Reaction-Diffusion Equations: Homotopy Pertuburation Approach

Introduction

It is the purpose of this paper to present the approximate analytical and numerical solution of cubic autocatalytic reaction-diffusion equations. There is a great deal of interest in chemical reactions which exhibit oscillatory solutions. These oscillations occur due to feedback in the system either chemical feedback such as autocatalysis or temperature feedback due to non-isothermal reaction. Two well-studied examples of reactions exhibiting oscillatory solutions are the isothermal Belousov-Zhabotinsky reaction and Sal’nikov thermokinetic oscillator. However simple a chemical scheme we are seeking, it must have at least two independent variables, if it is to represent oscillations. Although a single independent variable can lead to multiple stationary states, and hence to ‘ignitions’ and ‘extinctions’, oscillatory behaviour requires a richer soil than multi stability. The simplest polynomial necessary is a cubic, and cubic autocatalysis as a source of multi stability in an isothermal open system was clearly, although briefly, expounded by Zel’dovich & Zysin in 1941. Accordingly the chemist wishing to invoke autocatalysis to explain oscillations should first look for the simplest manner of decoupling any stoichiometric connection between reactant and catalyst concentrations. If the autocatalysis by the product B has the form A+B→2B or A+2B→3B, then adding the reaction B→C is sufficient for this task. This simply means the catalyst has a finite lifetime, its activity here decaying in a first-order way. These are the reactions constituting the ‘autocatalator.’

The Gray-Scott scheme, which represents cubic-autocatalysis with linear catalyst decay, has been much considered, because of its decay, has been much considered, because of its multiple steady-state responses and oscillatory solutions. See Gray (1988) and Gray & Scott (1990) for reviews and descriptions of much of this work. The scheme is A+2B→3B, rate = βab², B→C, rate = β𝛾b, where the concentrations of the reactant and autocatalyst a and b, respectively. The parameters β and 𝛾 are rate constants. The catalyst is not stable, but undergoes a simple linear decay to a product C . This allows a much wider variety of behavior in the system than does the cubic reaction alone. This system of equations has been considered by Kay & Scott (1988) numerically. Recently Marchant (2002) obtained the steady-state solutions for the cubic-autocatalytic reaction with linear decay in a reaction-diffusion cell using semi-analytical method. However, to the best of our knowledge there was no analytical result corresponding to the steady-state concentration of reactant and autocatalyst for all positive values of parameters have reported. The purpose of this communication is to derive the approximate analytical expressions for the concentrations of the reactant and autocatalyst for steady and non steady state, using Homotopy perturbation method and Laplace transformation.