On a modifcation of Olver's method: a special case

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Date
2016Version
Acceso abierto / Sarbide irekia
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Artículo / Artikulua
Version
Versión aceptada / Onetsi den bertsioa
Project Identifier
ES/1PE/MTM2014-52859
Impact
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10.1007/s00365-015-9298-y
Abstract
We consider the asymptotic method designed by Olver (Asymptotics and
special functions. Academic Press, New York, 1974) for linear differential equations of
the second order containing a large (asymptotic) parameter : xm y −2 y = g(x)y,
with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, especially the
cases m = 0, ±1, giving the Poincaré-type asymptotic expansions of two indep ...
[++]
We consider the asymptotic method designed by Olver (Asymptotics and
special functions. Academic Press, New York, 1974) for linear differential equations of
the second order containing a large (asymptotic) parameter : xm y −2 y = g(x)y,
with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, especially the
cases m = 0, ±1, giving the Poincaré-type asymptotic expansions of two independent
solutions of the equation. The case m = 2 is different, as the behavior of the solutions
for large is not of exponential type, but of power type. In this case, Olver’s theory
does not give many details. We consider here the special case m = 2. We propose
two different techniques to handle the problem: (1) a modification of Olver’s method
that replaces the role of the exponential approximations by power approximations,
and (2) 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. [--]
Subject
Publisher
Springer US
Published in
Constructive Approximation (2016) 43:273–290
Description
This is a post-peer-review, pre-copyedit version of an article published in Constructive Approximation. The final authenticated version is available online at: https://doi.org/10.1007/s00365-015-9298-y
Departament
Universidad Pública de Navarra. Departamento de Ingeniería Matemática e Informática /
Nafarroako Unibertsitate Publikoa. Matematika eta Informatika Ingeniaritza Saila
Publisher version
Sponsorship
The Dirección General de Ciencia y Tecnología (REF.MTM2014-52859) is acknowledged for its financial support.