Asymptotic Expansion

Abstract

Asymptotic expansion is a method that approximately evaluates an integral or solves an initial and/or boundary value problem for ordinary or partial differential equations. Among these integrals or boundary value problems, there must be a small parameter. This small parameter characterizes the scales of the practical problem whose mathematical model is the integral or boundary value problem. An asymptotic expansion is a formal integer power or fractional power series expansion of the small parameter. The coefficient of the each power yields a simpler integral or equation, which sometimes can be solved analytically. This greatly reduces the complexity of the original problem. When many terms are taken, the expansion gets so messy that one can never carry out the expansion to many orders. Here, “many” usually means more than three or four. Fortunately, the formal series expansion as an asymptotic expansion is usually a divergent series, yet its partial sum of few terms usually represents a good approximation to the original quantity we would like to find. Here, “few” usually means two or three.