Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter

Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter

In previous papers 6-8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not considered in Olver"s theory Olver, 1974]. In this paper we go one step forward and consider linear differential equations of the third order: $y"+a\Lambda^2 y"+b\Lambda^3y=f(x)y"+g(x)y$, with $a,b\in\mathbb{C}$ fixed, $f"$ and $g$ continuous, and $\Lambda$ a large positive parameter. We propose two different techniques to handle the problem: (i) a generalization of Olver"s method 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. As an application of the theory, we obtain new convergent and asymptotic expansions of the Pearcey integral $P(x,y)$ for large $|x|$.