Convergent and asymptotic expansions of solutions of differential equations with a large parameter: Olver cases II and III
Fecha
2015Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión publicada / Argitaratu den bertsioa
Impacto
|
10.1216/JIE-2015-27-1-27
Resumen
This paper continues the investigation initiated in [Lopez, 2013]. We consider the asymptotic
method designed by F. Olver [Olver, 1974] for linear differential equations of the second
order containing a large (asymptotic) parameter . We consider here the second and third
cases studied by Olver: differential equations with a turning point (second case) or a singular
point (third case). It is ...
[++]
This paper continues the investigation initiated in [Lopez, 2013]. We consider the asymptotic
method designed by F. Olver [Olver, 1974] for linear differential equations of the second
order containing a large (asymptotic) parameter . We consider here the second and third
cases studied by Olver: differential equations with a turning point (second case) or a singular
point (third case). It is well-known that his method gives the Poincar´e-type asymptotic
expansion of two independent solutions of the equation in inverse powers of . In this paper
we add initial conditions to the differential equation and consider the corresponding initial
value problem. By using the Green function of an auxiliary problem, we transform the initial
value problem into a Volterra integral equation of the second kind. Then, using a fixed point
theorem, we construct a sequence of functions that converges to the unique solution of the
problem. This sequence has also the property of being an asymptotic expansion for large
(not of Poincar´e-type) of the solution of the problem. Moreover, we show [--]
Materias
Second order differential equations. Turning points. Regular singular points. Volterra integral equations of the second kind. Asymptotic expansions. Green functions. Fixed point theorems. Airy functions. Bessel functions.
Editor
Rocky Mountain Mathematics Consortium
Publicado en
Journal of Integral Equations and Applications Volume 27, Number 1 (2015), 27-45.
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. MTM2010-21037) is acknowledged for its financial support