A computational method for singularly perturbed nonlinear differential-difference equations with small shift

This paper is devoted to the numerical study of the boundary value problems for nonlinear singularly perturbed differential-difference equations with small delay. Quasilinearization process is used to linearize the nonlinear differential equation. After applying the quasilinearization process to the nonlinear problem, a sequence of linearized problems is obtained. To obtain parameter-uniform convergence, a piecewise-uniform mesh is used, which is dense in the boundary layer region and coarse in the outer region. The parameter-uniform convergence analysis of the method has been discussed. The method has shown to have almost second-order parameter-uniform convergence. The effect of small shift on the boundary layer(s) has also been discussed. To demonstrate the performance of the proposed scheme two examples have been carried out. The maximum absolute errors and uniform rates of convergence have been presented in the form of the tables. (C) 2009 Elsevier Inc. All rights reserved.