Received 29.07.2024, Revised 25.10.2024, Accepted 22.11.2024

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

Zh.K. Turkmanov, M. Karynbaeva

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

asymptotic expansions, nonlinear equations, ordinary differential equations, small parameter, singular perturbations, boundary layers, parameterization, asymptotic analysis, solution behavior, Lighthill, Prandtl-Tikhonov
229-236
Turkmanov, Zh.K., & Karynbaeva, M. (2024). ASYMPTOTIC EXPANSIONS OF SOLUTIONS TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH A SMALL PARAMETER. Bulletin of the Bishkek State University, 22(3), 229-236. https://doi.org/10.35254/bsu/2024.69.35

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