On the analytic solution of the Cauchy problem
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- by Xiang-dong Hou
- Proc. Amer. Math. Soc. 137 (2009), 597-606
- DOI: https://doi.org/10.1090/S0002-9939-08-09493-8
- Published electronically: August 22, 2008
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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
- F. Faà di Bruno, Note sur un nouvelle formule de calcul différentiel, Quarterly Journal of Pure and Applied Mathematics 1 (1857), 359–360.
- G. M. Constantine and T. H. Savits, A multivariate Faà di Bruno formula with applications, Trans. Amer. Math. Soc. 348 (1996), no. 2, 503–520. MR 1325915, DOI 10.1090/S0002-9947-96-01501-2
- Alex D. D. Craik, Prehistory of Faà di Bruno’s formula, Amer. Math. Monthly 112 (2005), no. 2, 119–130. MR 2121322, DOI 10.2307/30037410
- Wolfgang Gröbner, Die Lie-Reihen und ihre Anwendungen, Mathematische Monographien, vol. 3, VEB Deutscher Verlag der Wissenschaften, Berlin, 1967 (German). Zweite, überarbeitete und erweiterte Auflage. MR 0217392
- O. M. Gvozdetskiĭ and V. P. Igumnov, Representation of solutions of ordinary differential equations in the form of Lie series, Ukrain. Mat. Zh. 38 (1986), no. 2, 218–220, 269 (Russian). MR 841056
- Michael Hardy, Combinatorics of partial derivatives, Electron. J. Combin. 13 (2006), no. 1, Research Paper 1, 13. MR 2200529, DOI 10.37236/1027
- V. P. Igumnov, Representation of solutions of differential equations by modified Lie series, Differentsial′nye Uravneniya 20 (1984), no. 6, 952–958 (Russian). MR 751841
- 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), no. 1754, 983–1019. MR 1694700, DOI 10.1098/rsta.1999.0362
- Nathan Jacobson, Lectures in abstract algebra. Vol. I, Graduate Texts in Mathematics, No. 30, Springer-Verlag, New York-Heidelberg, 1975. Basic concepts; Reprint of the 1951 edition. MR 0392227
- Warren P. Johnson, The curious history of Faà di Bruno’s formula, Amer. Math. Monthly 109 (2002), no. 3, 217–234. MR 1903577, DOI 10.2307/2695352
- Wilhelm Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math. 7 (1954), 649–673. MR 67873, DOI 10.1002/cpa.3160070404
- R. Most, Ueber die höheren differentialquotienten, Mathematische Annalen 4 (1871), 499–504.
- Ivan Niven, Formal power series, Amer. Math. Monthly 76 (1969), 871–889. MR 252386, DOI 10.2307/2317940
Bibliographic Information
- Xiang-dong Hou
- Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620
- Email: xhou@math.usf.edu
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2448581