Abstract:
Uniform asymptotics is found for a solution of the initial value problem to the
equation $ \varepsilon^2 u '= -u^2 + \varepsilon f (x) $, singularly depending on a small parameter $\varepsilon$. Equations of this type are already well studied, but this
equation represents an unexplored case of the right-hand side behavior.
By the method of asymptotics matching the three-scale asymptotic expansion for a solution is constructed and is justificated by the method of upper and lower solutions.
Keywords:asymptotic expansion, small parameter, initial value problem,
asymptotics matching method, intermediate expansion, Riccati equation.