Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter
Ver/
Fecha
2018Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión publicada / Argitaratu den bertsioa
Impacto
|
10.11948/2018.965
Resumen
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′ ...
[++]
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|. [--]
Materias
Third-order differential equations,
Asymptotic expansions,
Green’s functions,
Banach’s fixed point theorem,
Pearcey integral
Editor
Shanghai Normal University Wilmington Scientific Publisher
Publicado en
Journal of Applied Analysis and Computation, Volume 8, Number 3, June 2018
Departamento
Universidad Pública de Navarra. Departamento de Ingeniería Matemática e Informática /
Nafarroako Unibertsitate Publikoa. Matematika eta Informatika Ingeniaritza Saila
Versión del editor
Entidades Financiadoras
The Ministerio de Econom´ıa y Competitividad (REF. MTM2014-52859-P) is acknowledged by its financial support.