On a modifcation of Olver's method: a special case
Ver/
Fecha
2016Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión aceptada / Onetsi den bertsioa
Impacto
|
10.1007/s00365-015-9298-y
Resumen
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 i ...
[++]
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. [--]
Materias
Second-order differential equations,
Asymptotic expansions,
Green’s functions,
Banach’s fixed point theorem
Editor
Springer US
Publicado en
Constructive Approximation (2016) 43:273–290
Notas
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
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 Dirección General de Ciencia y Tecnología (REF.MTM2014-52859) is acknowledged for its financial support.