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

Formal and convergent solutions of singular partial differential equations

  • Gunter Bengel
  • Raymond Gérard


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.


Differential Equation Partial Differential Equation Linear Equation Differential Operator Nonlinear Equation 
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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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