Numerical solutions of a nonlinear reaction-diffusion system

Abstract

This paper is concerned with finite difference solutions of a coupled system of nonlinear reaction-diffusion equations. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. The existence and uniqueness of a non-negative finite difference solution and three monotone iterative algorithms for the computation of the solutions are given. It is shown that the time-dependent problem has a unique non-negative solution, whereas the steady-state problem may have multiple non-negative solutions depending on the parameters in the problem. The different non-negative steady-state solutions can be computed from the monotone iterative algorithms by choosing different initial iterations. Also discussed is the asymptotic behaviour of the time-dependent solution in relation to the steady-state solutions. The asymptotic behaviour result gives some conditions ensuring the convergence of the time-dependent solution to a positive or semitrivial non-negative steady-state solution. Numerical results are given to demonstrate the theoretical analysis results.

Keywords:

• reaction-diffusion system
• finite difference solution
• monotone iterations
• asymptotic behaviour
• upper and lower solutions

2000 AMS Mathematics Subject Classifications :

• 65M06
• 65N06
• 65N22

Computing Classification System :

• G.1.8

Acknowledgements

The author would like to thank the referees for valuable comments and suggestions that helped in improving the presentation of this paper. This work was supported in part by the National Natural Science Foundation of China No. 10571059, E-Institutes of Shanghai Municipal Education Commission No. E03004, and Shanghai Leading Academic Discipline Project No. B407.