We consider a nonlinear Laplace equation Δu = f(x,u) in two variables. Following the methods of B. Braaksma [Br] and J. Ecalle used for some nonlinear ordinary differential equations we construct first a formal power series solution and then we prove the convergence of the series in the same class as the function f in x.

We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation ${\partial }_{t}u-{\partial }_x²u={\partial }_x\left(u²\right)$ to be convergent or Borel summable.