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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
DOI: https://doi.org/10.1090/S0002-9939-08-09493-8
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.


References [Enhancements On Off] (What's this?)

  • 1. F. Faà di Bruno, Note sur un nouvelle formule de calcul différentiel, Quarterly Journal of Pure and Applied Mathematics 1 (1857), 359-360.
  • 2. G. M. Constantine and T. H. Savits, A multivariate Faà di Bruno formula with applications, Trans. Amer. Math. Soc. 348 (1996), 503-520. MR 1325915 (96g:05008)
  • 3. A. D. D. Craik, Prehistory of Faá di Bruno's formula, Amer. Math. Monthly 112 (2005), 119-130. MR 2121322 (2006a:40001)
  • 4. W. Gröbner, Die Lie-Reihen und ihre Anwendungen, Deutscher Verlag der Wissenschaften, Berlin, 1967. MR 0217392 (36:482)
  • 5. O. M. Gvozdetskiĭand V. P. Igumnov, Representation of solutions of ordinary differential equations in the form of Lie series, Ukrainian Math. J. 38 (1986), 192-194 (English translation of Ukrain. Mat. Zh.). MR 841056 (87e:34011)
  • 6. M. Hardy, Combinatorics of partial derivatives, Electron. J. Combin. 13 (2006), #R1. MR 2200529 (2007d:26007)
  • 7. V. P. Igumnov, Representation of solutions of differential equations by modified Lie series, Differential Equations 20 (1984), 688-694 (English translation of Differentsial$ ^\prime$nye Uravneniya). MR 751841 (86a:34016)
  • 8. A. Iserles and S. P. Nørsett, On the solution of linear differential equations in Lie groups, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999), 983-1019. MR 1694700 (2000d:34022)
  • 9. N. Jacobson, Lectures in Abstract Algebra. Vol. I, Springer-Verlag, New York-Heidelberg, 1975, p. 19. MR 0392227 (52:13044)
  • 10. W. P. Johnson, The curious history of Faà di Bruno's formula, Amer. Math. Monthly 109 (2002), 217-234. MR 1903577 (2003d:01019)
  • 11. W. Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math. 7 (1954), 649-673. MR 0067873 (16:790a)
  • 12. R. Most, Ueber die höheren differentialquotienten, Mathematische Annalen 4 (1871), 499-504.
  • 13. I. Niven, Formal power series, Amer. Math. Monthly 76 (1969), 871-889. MR 0252386 (40:5606)

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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: https://doi.org/10.1090/S0002-9939-08-09493-8
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.

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