‘Shooting method’ for singularly perturbed one-dimensional reaction-diffusion neumann problems

Abstract

A numerical method for singularly perturbed two-point boundary-value problems for second-order ordinary differential equations subject to Neumann boundary conditions is proposed. In this method, the given interval (the domain of definition of the differential equation) is divided into one ‘outer region’ and two ‘inner regions’. Two initial-value problems associated with the inner region will be derived from the given boundary-value problem. One boundary-value problem derived from the given problem will correspond to the outer region. In each of the two inner regions an initial-value problem is solved by the fourth-order Runge-Kutta method. The boundary-value problem in the outer region is solved by a classical finite difference scheme. A combination of the solutions so obtained yields a numerical solution of the boundary-value problem on the whole interval. The implementation of the method on parallel architectures is discussed. Numerical examples are presented in support of the proposed method.

Keywords:

• Singular perturbation problem
• asymptotic expansion
• boundary layer
• numerical solution for ordinary differential equations
• parallel computation

C.R. Category:

• G.1.7