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manuscripta mathematica

, Volume 38, Issue 3, pp 343–373 | Cite as

Formal and convergent solutions of singular partial differential equations

  • Gunter Bengel
  • Raymond Gérard
Article

Abstract

We consider the problem of the existence of convergent series solutions for partial differential operators of the form\(P\left( {x,x\frac{\partial }{{\partial x}}} \right) = \sum\limits_{\left| \ell \right| \leqq d} {a_\ell \left( x \right)\left( {x\frac{\partial }{{\partial x}}} \right)^\ell } \). We give first conditions for P such that the linear equation Pu=f has an analytic solution, then we solve nonlinear equations of the form Pu=F(x,u). For applications we treat also cases with parameters and give a proof of a theorem of S. Kaplan [5]. In the last section we consider a case where small denominators occur.

Keywords

Differential Equation Partial Differential Equation Linear Equation Differential Operator Nonlinear Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Gunter Bengel
    • 1
  • Raymond Gérard
    • 2
  1. 1.Mathematisches InstitutWestfälische Wilhelms UniversitätMünsterBundesrepublik Deutschland
  2. 2.I.R.M.A.France

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