On a Modification of Olver’s Method: A Special Case

Ferreira, C. (Universidad de Zaragoza) ; López, J.L. ; Pérez Sinusía, E. (Universidad de Zaragoza)

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 (Formula presented.): (Formula presented.), with (Formula presented.) and g continuous. Olver studies in detail the cases (Formula presented.), especially the cases (Formula presented.), giving the Poincaré-type asymptotic expansions of two independent solutions of the equation. The case (Formula presented.) is different, as the behavior of the solutions for large (Formula presented.) 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 (Formula presented.). 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.

Idioma: Inglés

DOI: 10.1007/s00365-015-9298-y

Año: 2016

Publicado en: CONSTRUCTIVE APPROXIMATION 43, 2 (2016), 273–290
ISSN: 0176-4276

Factor impacto JCR: 0.964 (2016)

Categ. JCR: MATHEMATICS rank: 67 / 310 = 0.216 (2016) - Q1 - T1

Factor impacto SCIMAGO: 1.094 - Computational Mathematics (Q1) - Mathematics (miscellaneous) (Q1) - Analysis (Q2)

Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2010-21037

Tipo y forma: Article (PostPrint)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)