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Two-point Taylor expansions and one-dimensional boundary value problems

Authors: José L. López and Ester Pérez Sinusía
Journal: Math. Comp. 79 (2010), 2103-2115
MSC (2010): Primary 34A25, 34B05, 41A58
Published electronically: April 29, 2010
MathSciNet review: 2684357
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Abstract: We consider second-order linear differential equations $ \varphi(x)y''+f(x)y'+g(x)y=h(x)$ in the interval $ (-1,1)$ with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions. We consider $ \varphi(x)$, $ f(x)$, $ g(x)$ and $ h(x)$ analytic in a Cassini disk with foci at $ x=\pm 1$ containing the interval $ (-1,1)$. The two-point Taylor expansion of the solution $ y(x)$ at the extreme points $ \pm 1$ is used to give a criterion for the existence and uniqueness of solution of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the solution(s) when it exists.

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Additional Information

José L. López
Affiliation: Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006-Pamplona, Spain

Ester Pérez Sinusía
Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, 50018-Zaragoza, Spain

Keywords: Second-order linear differential equations, boundary value problem, Frobenius method, two-point Taylor expansions
Received by editor(s): May 5, 2009
Published electronically: April 29, 2010
Additional Notes: The Ministerio de Ciencia y Tecnología (REF. MTM2007-63772) and the Gobierno de Navarra (Res. 228/2008) are acknowledged by their financial support. The Department of Theoretical Physics of the University of Zaragoza is also acknowledged by its hospitality.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.