'Shooting method' for singularly perturbed one-dimensional reaction-diffusion Neumann problems

A numerical method for singularly perturbed two-point boundary-value proble ms for second-order ordinary differential equations subject to Neumann boun dary conditions is proposed. In this method, the given interval (the domain of definition of the differential equation) is divided into one 'outer reg ion' and two 'inner regions'. Two initial-value problems associated with th e inner region will be derived from the given boundary-value problem. One b oundary-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 co mbination of the solutions so obtained yields a numerical solution of the b oundary-value problem on the whole interval. The implementation of the meth od on parallel architectures is discussed. Numerical examples are presented in support of the proposed method.