manuscripta mathematica

, Volume 38, Issue 3, pp 343–373

# 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
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