A singular perturbation method. Part I

An approach to singular perturbation problems, introduced by Mahony [1] which arose out of the consideration of a problem involving a boundary layer is applicable to other singular perturbation problems. It lends itself particularly well to problems involving wave propagation, where “multiple scales” are involved. In this paper and the paper to follow, interest is centered around the equation \[ {\epsilon ^3}{\nabla ^2}\psi - g\left ( x \right )\psi = 0\], where $\epsilon$ is a small positive parameter and $g\left ( x \right )$ is a bounded function of $x$ which vanishes along simple closed curves in the solution domain. The one-dimensional case (the Langer turning point problem) is considered in this paper and it will be shown that the approach leads to exactly the same results as obtained by Langer and his associates using a “related equation” method.