Higher-Order Monotone Iterative Methods For Numerical Solutions Of Nonlinear Reaction-Diffusion Equations With Time Delay

This paper is concerned with the computational algorithms for finite difference systems of a class of nonlinear reaction-diffusion equation with time delay. A higher-order monotone iterative method is presented by using the method of upper and lower solutions. It is shown that the sequence of iterations converges monotonically to a unique solution of the system in a sector between a pair of upper and lower solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration. The rate of convergence of the iterations is estimated explicitly by the infinity norm. The order of convergence attains at p+2 where p≥1 is a positive integer depending on the construction of the iterative method. An application is given to an enzyme substrate reaction model with time delay. Some numerical results are presented to illustrate the effectiveness of the proposed method.