Asymptotic solutions of a fourth order differential equation

In this paper, we derive uniform asymptotic expansions of solutions to the fourth order differential equation y⁽⁴⁾ + λ ²(xy″ + y) = 0, where x is a real variable and λ is a large positive parameter. The solutions of this differential equation can be expressed in the form of contour integrals, and uniform asymptotic expansions are derived by using the cubic transformation introduced by Chester, Friedman, and Ursell in 1957 and the integration-by-part technique suggested by Bleistein in 1966. There are two advantages to this approach: (i) the coefficients in the expansion are defined recursively, and (ii) the remainder is given explicitly. Moreover, by using a recent method of Olde Daalhuis and Temme, we extend the validity of the uniform asymptotic expansions to include all real values of x. © 2007 by the Massachusetts Institute of Technology.