Asymptotic Initial Value Technique for singularly perturbed convection–diffusion delay problems with boundary and weak interior layers

Abstract

In this paper a numerical method named as Asymptotic Initial Value Technique (AIVT) is suggested to solve the singularly perturbed boundary value problem for the second order ordinary delay differential equation with the discontinuous convection–diffusion coefficient term. In this technique, the original problem of solving the second order differential equation is reduced to solving three first order differential equations, one of which is a delay differential equation and other two are singularly perturbed problems. The singularly perturbed problems are solved by the second order hybrid finite difference scheme, whereas the delay problem is solved by the fourth order Runge–Kutta method with Hermite interpolation. An error estimate is derived by using the supremum norm and it is of order O ( ε + N − 2 ln 2 N ) , where N and ε are the discretization parameter and the perturbation parameter, respectively. Numerical results are provided to illustrate the theoretical results.