Abstract
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:<i>y</i>''' + <i>a</i>Λ<sup>2</sup><i>y</i>' +<i>b</i>Λ<sup>3</sup><i>y</i>=<i>f</i>(<i>x</i>)<i>y</i>' +<i>g</i>(<i>x</i>)<i>y</i>, with <i>a, b</i> ∈ C fixed, <i>f</i>' and <i>g</i> continuous, and Λ 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 <i>P</i> (<i>x, y</i>) for large|<i>x</i>|.
Highlights
The most famous asymptotic method for second-order linear differential equations containing a large parameter is, no doubt, Olver’s method [14, Chaps. 10, 11, 12]
In [14, Chap. 10], Olver considers a differential equation without singular or transition points, an equation of the form y′′ − Λ2y = f (z)y, Λ → ∞, (1.1)
We propose the following representations in the form of formal asymptotic expansions for large Λ, Yj(z) = Yj,n(z) + Rj,n(z), j = 1, 2, 3, (4.1)
Summary
The most famous asymptotic method for second-order linear differential equations containing a large parameter is, no doubt, Olver’s method [14, Chaps. 10, 11, 12]. In this paper we go one step forward and propose a different generalization of Olver’s theory, we consider a third-order linear differential equation without singular or transition points. The purpose of this paper is to analyze the asymptotic behavior of the solutions of this equation for large Λ To this end, we use a fixed point theorem and the Green function of an auxiliary initial value problem to derive an asymptotic as well as convergent expansion of any solution of the equation in terms of iterated integrals of f (z) and g(z); this technique is based on our previous investigations [10].
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