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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the analytic solution of the Cauchy problem


Author: Xiang-dong Hou
Journal: Proc. Amer. Math. Soc. 137 (2009), 597-606
MSC (2000): Primary 34A25, 05A15
Published electronically: August 22, 2008
MathSciNet review: 2448581
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Abstract: Derivatives of a solution of an ODE Cauchy problem can be computed inductively using the Faà di Bruno formula. In this paper, we exhibit a noninductive formula for these derivatives. At the heart of this formula is a combinatorial problem, which is solved in this paper. We also give a more tractable form of the Magnus expansion for the solution of a homogeneous linear ODE.


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Additional Information

Xiang-dong Hou
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620
Email: xhou@math.usf.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09493-8
PII: S 0002-9939(08)09493-8
Keywords: Cauchy problem, ODE, Magnus expansion, partial order, combinatorics, Bruno's formula
Received by editor(s): April 13, 2007
Received by editor(s) in revised form: January 24, 2008
Published electronically: August 22, 2008
Communicated by: Jim Haglund
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.