ASYMPTOTIC EXPANSIONS OF SOLUTIONS TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH A SMALL PARAMETER

This study examines the asymptotic expansions of solutions to nonlinear ordinary differential equations with a small parameter. Two classes of singularly perturbed equations are considered: Prandtl-Tikhonov type equations characterized by boundary layers and stable solutions with small parameters, and Lighthill-type equations distinguished by internal layers and more complex solution structures. A parameterization method was presented to analyze the solution behavior up to a singular point. The existence of a solution to the Cauchy problem for the Lighthill equation under certain conditions was proven. The advantage of the parameterization method over the classical small-parameter method in constructing asymptotic expansions of solutions near a singular point was demonstrated.