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SPRING 2015 Convergent and asymptotic expansions of solutions of differential equations with a large parameter: Olver cases II and III
Chelo Ferreira, José L. López, Ester Pérez Sinusía
J. Integral Equations Applications 27(1): 27-45 (SPRING 2015). DOI: 10.1216/JIE-2015-27-1-27

Abstract

We consider the asymptotic method designed by Olver \cite{olver} for linear differential equations of second order containing a large (asymptotic) parameter $\Lambda$, in particular, 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 $\Lambda$. In this paper, we add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's 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 also has the property of being an asymptotic expansion for large $\Lambda$ (not of Poincar\'e-type) of the solution of the problem. Moreover, we show that the technique also works for nonlinear differential equations with a large parameter.

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Chelo Ferreira. José L. López. Ester Pérez Sinusía. "Convergent and asymptotic expansions of solutions of differential equations with a large parameter: Olver cases II and III." J. Integral Equations Applications 27 (1) 27 - 45, SPRING 2015. https://doi.org/10.1216/JIE-2015-27-1-27

Information

Published: SPRING 2015
First available in Project Euclid: 24 February 2015

zbMATH: 1337.34057
MathSciNet: MR3316977
Digital Object Identifier: 10.1216/JIE-2015-27-1-27

Subjects:
Primary: 34A12, 34B27, 41A58, 41A60, 45D05

Rights: Copyright © 2015 Rocky Mountain Mathematics Consortium

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Vol.27 • No. 1 • SPRING 2015
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