Approximation methods for solving the Cauchy problem

Mortici, Cristinel. "Approximation methods for solving the Cauchy problem." Czechoslovak Mathematical Journal 55.3 (2005): 709-718. <http://eudml.org/doc/30981>.

@article{Mortici2005,
abstract = {In this paper we give some new results concerning solvability of the 1-dimensional differential equation $y^\{\prime \} = f(x,y)$ with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if $f$ is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.},
author = {Mortici, Cristinel},
journal = {Czechoslovak Mathematical Journal},
keywords = {Cauchy problem; Lipschitz function; Picard theorem; succesive approximations method; contractions principle; Cauchy problem; Lipschitz function; Picard theorem; succesive approximations method; contractions principle},
language = {eng},
number = {3},
pages = {709-718},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Approximation methods for solving the Cauchy problem},
url = {http://eudml.org/doc/30981},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Mortici, Cristinel
TI - Approximation methods for solving the Cauchy problem
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 3
SP - 709
EP - 718
AB - In this paper we give some new results concerning solvability of the 1-dimensional differential equation $y^{\prime } = f(x,y)$ with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if $f$ is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.
LA - eng
KW - Cauchy problem; Lipschitz function; Picard theorem; succesive approximations method; contractions principle; Cauchy problem; Lipschitz function; Picard theorem; succesive approximations method; contractions principle
UR - http://eudml.org/doc/30981
ER -