Renormalization Group Method for Singular Perturbation Initial Value Problems with Delays

The aim of this paper is to investigate the initial value problems of a class of singularly perturbed differential equations with two delays. First, we deal with the case of two commensurate delays, which is different from the case of a single delay. We carry out a suitable division of time interval and combine the renormalization group method to construct the effective approximate solution, then give the rigorous error estimate. Second, for the case of two incommensurate delays, in light of the approximate auxiliary systems with two commensurate delays, we obtain the approximate solution and analyze the error estimate. Finally, some numerical comparisons among the exact solution and approximate solutions by averaging method, multiple scale method, renormalization group method are given to illustrate the validness of our results.