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

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    Date
    2016
    Author
    Ferreira, Chelo 
    López García, José Luis 
    Pérez Sinusía, Ester 
    Version
    Acceso abierto / Sarbide irekia
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    Artículo / Artikulua
    Version
    Versión aceptada / Onetsi den bertsioa
    Project Identifier
    ES/1PE/MTM2014-52859 
    Impact
     
     
     
    10.1007/s00365-015-9298-y
     
     
     
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    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
    Second-order differential equations, Asymptotic expansions, Green’s functions, Banach’s fixed point theorem
     
    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
    https://doi.org/10.1007/s00365-015-9298-y
    URI
    https://hdl.handle.net/2454/31770
    Sponsorship
    The Dirección General de Ciencia y Tecnología (REF.MTM2014-52859) is acknowledged for its financial support.
    Appears in Collections
    • Artículos de revista DIMI - MIIS Aldizkari artikuluak [43]
    • Artículos de revista - Aldizkari artikuluak [2160]
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