Uniformly convergent difference schemes for singularly perturbed initial-value problems

In this study, we investigate the numerical solutions of the linear first- and second-order singularly perturbed initial-value problems. Firstly, asymptotic estimates for these problems are established. Next, by method of integral identities using appropriate interpolating quadrature rules with the weight and remainder terms in integral form, the numerical methods are constructed. The difference schemes are shown to converge to the continuous solutions uniformly with respect to the perturbation parameter. Also the problem is tested on concrete examples and it is shown that numerical results are essentially in agreement with the theoritical analysis.