A Parameter Uniform Almost First Order Convergent Numerical Method for a Semi-Linear System of Singularly Perturbed Delay Differential Equations

In this paper an initial value problem for asemi-linear system of two singularly perturbed first order delay differential equations is considered on the interval(0,2]. The components of the solution of this system exhibit initial layers at 0 and interior layers at 1. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first order convergent in the maximum norm uniformly in the perturbation parameters

AN INITIAL VALUE TECHNIQUE FOR SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEMS WITH A NEGATIVE SHIFT

Abstract. In this paper, a numerical method named as Initial Value Technique (IVT) is suggested to solve singularly perturbed boundary value problems for second order ordinary differential equations of reactiondiffusion type with a delay (negative shift). In this technique, the original problem of solving the second order differential equation is reduced to solving four first order singularly perturbed differential equations without delay and one algebraic equation with a delay. The singularly perturbed problems are solved by a second order hybrid finite difference scheme. An error estimate is derived by using supremum norm and it is of order O(ε + N −2 ln 2 N ), where N is a discretization parameter and ε is the perturbation parameter. Numerical results are provided to illustrate the theoretical results