Details

The governing equations are

and

.

With the membrane at , the initial conditions are

,

and , with boundary conditions

at and . Here and are the diffusion coefficients of and , respectively, is distance, and is time.

The system can be simplified by making the transformations

and

to obtain the system

,

,

which has the following analytical solution when [1]:

,

,

with

, where is a free parameter taken as .

Reference

[1] A. H. Salas, L. J. Martinez H., and O. Fernandez S., "Reaction-Diffusion Equations: A Chemical Application," Scientia et Technica, 17(46), 2010 pp. 134–137. www.redalyc.org/pdf/849/84920977041.pdf.