Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables

Maria Ewa Pliś; Bogdan Ziemian

Annales Polonici Mathematici (1997)

  • Volume: 67, Issue: 1, page 31-41
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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Maria Ewa Pliś, and Bogdan Ziemian. "Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables." Annales Polonici Mathematici 67.1 (1997): 31-41. <http://eudml.org/doc/270185>.

@article{MariaEwaPliś1997,
abstract = {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.},
author = {Maria Ewa Pliś, Bogdan Ziemian},
journal = {Annales Polonici Mathematici},
keywords = {Borel resummation; formal solutions; Laplace equation; Laplace transform; formal power series solution},
language = {eng},
number = {1},
pages = {31-41},
title = {Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables},
url = {http://eudml.org/doc/270185},
volume = {67},
year = {1997},
}

TY - JOUR
AU - Maria Ewa Pliś
AU - Bogdan Ziemian
TI - Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables
JO - Annales Polonici Mathematici
PY - 1997
VL - 67
IS - 1
SP - 31
EP - 41
AB - 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.
LA - eng
KW - Borel resummation; formal solutions; Laplace equation; Laplace transform; formal power series solution
UR - http://eudml.org/doc/270185
ER -

References

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  1. [Br] B. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier (Grenoble) 42 (1992), 517-541. 
  2. [Sz-Zie] Z. Szmydt and B. Ziemian, The Mellin Transformation and Fuchsian Type PDEs, Kluwer, 1992. 
  3. [Zie1] B. Ziemian, Generalized analytic functions with applications to singular ordinary and partial differential equations, Dissertationes Math. 354 (1996). 
  4. [Zie2] B. Ziemian, Leray residue formula and asymptotics of solutions to constant coefficient PDEs, Topol. Methods Nonlinear Anal. 3 (1994), 257-293. 

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