Application of the Two-Scale Expansion Method to First-Order Singularly Perturbed Problems

This article presents the theory and algorithm for applying the two-scale expansion method (method of multiple scales) to first-order singularly perturbed differential problems. The structure of the boundary layer arising due to the presence of a small parameter ε multiplying the highest-order derivative is investigated. Asymptotic expansions of the outer (slow-scale) and inner (fast-scale) solutions are constructed, and a composite solution is obtained using the matching principle. Error estimation is theoretically justified and verified through numerical experiments implemented in Python. Theoretical and numerical results confirm that the width of the boundary layer is of order δ ~ ε.