Convergent and asymptotic expansions of solutions of differential equations with a large parameter: Olver cases II and III

Abstract

We consider the asymptotic method designed by Olver \cite{olver} for linear differential equations of second order containing a large (asymptotic) parameter $\Lambda$, in particular, the second and third cases studied by Olver: differential equations with a turning point (second case) or a singular point (third case). It is well known that his method gives the Poincar\'e-type asymptotic expansion of two independent solutions of the equation in inverse powers of $\Lambda$. In this paper, we add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then, using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence also has the property of being an asymptotic expansion for large $\Lambda$ (not of Poincar\'e-type) of the solution of the problem. Moreover, we show that the technique also works for nonlinear differential equations with a large parameter.