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

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Shanghai Normal University
Wilmington Scientific Publisher
Acceso abierto / Sarbide irekia
Artículo / Artikulua
Versión publicada / Argitaratu den bertsioa

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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 ′′′ +aΛ2y′ +bΛ3y = f(x)y′ +g(x)y, with a, b ∈ C fixed, f′ and g 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 P(x, y) for large |x|.


Third-order differential equations, Asymptotic expansions, Green’s functions, Banach’s fixed point theorem, Pearcey integral


Ingeniería Matemática e Informática / Matematika eta Informatika Ingeniaritza



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The Ministerio de Econom´ıa y Competitividad (REF. MTM2014-52859-P) is acknowledged by its financial support.

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