Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions

Abstract

<p style='text-indent:20px;'>The approximate solutions of a two-cell reaction-diffusion model equation subjected to the Dirichlet conditions are obtained. The reaction is assumed to occur in the presence of cubic autocatalyst which decays to an inert compound in the first cell. Coupling with the reactant is assumed to be cubic in the concentrations. A linear exchange in the concentration of the reactant is taken between the two cells. The formal exact solution is found analytically. Here, in this investigation, use is made of the Picard iterative scheme which is constructed and applied after the exact one. The results obtained are compared with those found by means of a numerical method. It is observed that the solution obtained here is symmetric with respect to the mid-point of the container.The travelling wave is expected due to the parity of the space operator and the symmetric boundary conditions. Symmetric patterns, including among them a parabolic one, are observed for a large time.</p><p style='text-indent:20px;'>When the initial conditions are periodic, the most dominant modes travel at a constant speed for a large time. This phenomenon is highly affected by the rate of decay of the autocatalyst to an inert compound. The present work is of remarkably significant interest in chemical engineering as well as in other physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product, which is carried by controlling the initial concentration of the reactant. Furthermore, in the last section on conclusions, we have cited many potentially useful recent works related to the subject-matter of this investigation in order to provide incentive and motivation for making further advances by using space-time fractional derivatives along the lines of the problem of finding approximate analytical solutions of the reaction-diffusion model equations which we have discussed in this article.</p>