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Transactions of the American Mathematical Society

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Formal and convergent power series solutions of singular partial differential equations


Author: Stanley Kaplan
Journal: Trans. Amer. Math. Soc. 256 (1979), 163-183
MSC: Primary 35C10; Secondary 14B12, 32B05, 35A35
DOI: https://doi.org/10.1090/S0002-9947-1979-0546913-5
MathSciNet review: 546913
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Abstract: A class of singular first-order partial differential equations is described for which an analogue of a theorem of M. Artin on the solutions of analytic equations holds: given any formal power series solution and any nonnegative integer v, a convergent power series solution may be found which agrees with the given formal solution up to all terms of order $ \leqslant v$.


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

  • [1] M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291. MR 0232018, https://doi.org/10.1007/BF01389777
  • [2] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745
  • [3] F. R. Gantmacher, The theory of matrices. Vol. 2, Chelsea, New York.
  • [4] E. Goursat, Leçons sur l'intégration des équations aux dérivées partielles du premier ordre, A. Hermann, Librairie Scientifique, Paris, 1891.
  • [5] H. Poincaré, Sur les propriétés des fonctions définies par les équations aux différences partielles, Oeuvres, Tome 1. Gauthier-Villars, Paris, 1951.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0546913-5
Article copyright: © Copyright 1979 American Mathematical Society