Raf. J. of Comp. & Math’s. , Vol. 9, No. 2, 2012 243 Numerical Solution of a Reaction-Diffusion System with Fast Reversible Reaction by Using Adomian’s Decomposition Method and He’s Variational Iteration Method Ann J. Al-Sawoor Mohammed O. Al-Amr College of Computer Sciences and Mathematics University of Mosul Received on: 13/06/2012 Accepted on: 18/09/2012 ﺍﻟﻤﻠ ﺨ ﺹ    ﺍﻟﺒﺤﺙ ﻫﺫﺍ ﻓﻲ،  ﺘﻡ ﺇ ﺍﻟﺘﻔﺎﻋل ﻟﻨﻅﺎﻡ ﺍﻟﺘﻘﺭﻴﺒﻲ ﺍﻟﺤل ﻴﺠﺎﺩ- ﺒﺎﺴﺘﺨﺩﺍﻡ ﺴﺭﻴﻊ ﻋﻜﺴﻲ ﺘﻔﺎﻋل ﻤﻊ ﺍﻻﻨﺘﺸﺎﺭ ﻭﻫﻤﺎ ﹰﻤﺅﺨﺭﺍ ﺘﻁﻭﻴﺭﻫﺎ ﺘﻡ ﺍﻟﺘﻲ ﺍﻟﻔﻌﺎﻟﺔ ﺍﻟﻁﺭﻕ ﻤﻥ ﻁﺭﻴﻘﺘﻴﻥﺍﺩﻭﻤﻴﻥ ﺘﺤﻠﻴل ﻁﺭﻴﻘﺔ  (ADM) ﺍﻟﺘﻜﺭﺍﺭ ﻭﻁﺭﻴﻘﺔ  ﺍﻟﻤﺘﻐﺎﻴﺭ(VIM) . ﺇﻥﺘﺘﻁﻠﺏ ﺍﻟﻤﺘﻐﺎﻴﺭﺍﻟﺘﻜﺭﺍﺭﻁﺭﻴﻘﺔ ﺇﻴﺠﺎﺩﻻﻜﺭﺍﻨﻤﻀﺭﻭﺏ ﺞ، ﺘﺘﻁﻠﺏﺍﺩﻭﻤﻴﻥﺘﺤﻠﻴلﻁﺭﻴﻘﺔ ﺒﻴﻨﻤﺎ ﺇﻴﺠﺎﺩ ﺤﺩﻭﺩ ﻤﺘﻌﺩﺩﺍﺕ ﺍ ﺩﻭﻤﻴﻥ. ﻭ ﺍﻟﺘﻘﺭﻴﺒﻲ ﺍﻟﺤل ﺴﻠﻭﻙ ﺘﻭﻀﻴﺢ ﺘﻡ ﻜﻤﺎ ﺍﻟ ﺘﺄﺜﻴﺭﺍﺕ ﻟ ﻟـ ﻤﺨﺘﻠﻔﺔ ﻘﻴﻡt  ﻁﺭﻴﻕ ﻋﻥ ﺍﻟﺭﺴﻡ.  ABSTRACT In this paper, the approximate solution of a reaction-diffusion system with fast reversible reaction is obtained by using Adomian decomposition method (ADM) and variational iteration method (VIM) which are two powerful methods that were recently developed. The VIM requires the evaluation of the Lagrange multiplier, whereas ADM requires the evaluation of the Adomian polynomials. The behavior of the approximate solutions and the effects of different values of t are shown graphically. Keywords: reaction-diffusion system, fast reversible reaction, Adomian decomposition method, variational iteration method. 1. Introduction: Non-linear phenomena, that appear in many areas of scientific fields such as solid state physics, plasma physics, fluid dynamics, mathematical biology and chemical kinetics can be modeled by partial differential equation. A broad class of analytical solutions methods and numerical solutions methods were used in handle these problems [10]. In the 1980’s, Adomian [2–4] introduced a new powerful method which provides an efficient means for the analytic and numerical solution of differential equations. It is free from rounding off errors since, it does not involve discretization and is computationally inexpensive. Recently, the variational iteration method, which was first proposed by He [14– 17] in 1997, has many merits and advantages over the Adomian method. A comparative study between the two methods was conducted by Wazwaz [24]. The main advantage of the two methods is that it can be applied directly for all types of differential and integral equations, homogeneous or inhomogeneous. Another important advantage is that the methods are capable of greatly reducing the size of computational work, while still maintaining high accuracy of the numerical solution [24].