Dear Ariful Islam Arif,

You can find the following links and file:

http://pubs.acs.org/doi/abs/10.1021/j100110a017?journalCode=jpchax

http://www.scirp.org/journal/PaperInformation.aspx?paperID=38978

Dear Ariful, I found useful some papers by Prof. Alfio Quarteroni (RG member) on subject, as attached paper on Burger's equation See also A. Bayliss and others, An adaptive pseudo-spectral method for reaction diffusion problems.  Gianluca

See also the following link in RG Questions, about reaction-diffusion equation.  Gianluca

https://www.researchgate.net/post/Any_suggestions_for_the_discretization_of_coupled_reaction_diffusion_equations

Dear Ariful Islam Arif,

You can find the following links and file:

http://pubs.acs.org/doi/abs/10.1021/j100110a017?journalCode=jpchax

http://www.scirp.org/journal/PaperInformation.aspx?paperID=38978

I think This is best article yet written on solving RD equations using spectral method. They used ETD for time stepping which gives exponential convergence rate.

Solving reaction–diffusion equations 10 times faster Aly-Khan Kassam 1 Oxford University Computing Laboratory, Wolfson Bldg., Parks Road, Oxford OX1 3QD, UK.