Numerical approximations for singularly perturbed differential-difference BVPs with layer and oscillatory behavior

摘要

A numerical approximation for singularly perturbed linear second order boundary value problem with small shift (depending on a small parameter) in the convection term is considered. When the delay argument is sufficiently small say of o(e), to tackle the delay term, we have used Taylor's series expansion and presented a numerical approach to solve such type of boundary value problem. But in the case when the delay is of not sufficiently small the approach of simply expanding the shift term in Taylor's series and truncating may lead to misleading results, this is the motivation for this work. In this paper, we present a numerical scheme for solving such type of boundary value problems, which works nicely when delay argument is of O(ε). To handle the delay argument, we construct a special type of mesh so that the term containing delay lies on nodal points after discretization. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations. Comparisons of the numerical solutions are performed with standard upwind finite difference scheme on a special type of mesh to demonstrate the efficiency of the method.