Olver's asymptotic method: a special case

TL;DR

This paper examines Olver's asymptotic method for second-order linear differential equations with a large parameter, focusing on the special case where the parameter's power is 2, and introduces new techniques for this scenario.

Contribution

It proposes two novel approaches to analyze the case m=2, extending Olver's method and applying fixed point techniques to nonlinear equations with large parameters.

Findings

Modified Olver's method using power approximations.

Transformation into fixed point problem for solution construction.

Extension of techniques to nonlinear differential equations.

Abstract

We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equations of the second order containing a large (asymptotic) parameter $\Lambda$: $x^my"-\Lambda^2y=g(x)y$, with $m\in\mathbb{Z}$ and $g$ continuous. Olver studies in detail the cases $m\ne 2$, specially the cases $m=0,\pm 1$, giving the Poincar\'e-type asymptotic expansion of two independent solutions of the equation. The case $m=2$ is different, as the behavior of the solutions for large $\Lambda$ is not of exponential type, but of power type. In this case, Olver's theory does not give as many details as it gives in the cases $m\ne 2$. Then, we consider here the special case $m=2$. We propose two different techniques to handle the problem: (i) a modification of Olver's method that replaces the role of the exponential approximations by power approximations and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.