Picard–Lindelöf theorem

Summary

In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. Theorem Let D \subseteq \R \times \R^n be a closed rectangle with (t_0, y_0) \in \operatorname{int} D, the interior of D. Let f: D \to \R^n be a function that is continuous in t and Lipschitz continuous in y. Then, there exists some ε > 0 such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]. Proof sketch The proof relies on transforming the differential

Official source

https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (6)

Related publications

Related people

Related units

Related concepts (10)

Related concepts

Related courses (23)

Le cours donne une introduction à la théorie des EDO, y compris existence de solutions locales/globales, comportement asymptotique, étude de la stabilité de points stationnaires et applications, en particulier aux systèmes dynamiques et en biologie.

This course presents numerical methods for the solution of mathematical problems such as systems of linear and non-linear equations, functions approximation, integration and differentiation and differential equations.

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

Related courses

Related lectures (52)

Related lectures

The Adomian Decomposition Method for Nonlinear Partial Differential Equations

Natalie Sauerwald

The goal of this report is to study the method introduced by Adomian known as the Adomian Decomposition Method (ADM), which is used to find an approximate solution to nonlinear partial differential equations (PDEs) as a series expansion involving the recursive solution of linear PDEs. We first describe the method, giving two specific examples with different nonlinearities and show exactly how the method works for these problems. Some analytical convergence results are then given, along with numerical solutions to the examples demonstrating these convergence results. A discussion of parameters inside of these nonlinearities follows, both for polynomial nonlinearities and for the more complicated hyperbolic sine nonlinearity problem. Finally, we compare the ADM with the Picard method, pointing out some important differences and demonstrating them by solving the given examples with both methods and comparing the results.

2013

STOCHASTIC PROCESSES AND CONSTRUCTION OF BROWNIAN MOTION

Christophe-Aschkan Mery

This project offers a rigorous introduction to the tools needed to construct a continuous stochastic process. Among other things, we give a very detailed proof of the Kolmogorov continuity criterion. We then construct a Brownian Motion following the formalism of D. Revuz and M. Yor. That is, we see the BM as a linear isometry from a Hilbert space into a Gaussian space.

EPFL2013

Sur quelques inclusions différentielles de premier et de deuxième ordre

Gisella Croce

The principal subject of this thesis is the study of some Dirichlet problems related to differential inclusions of first and second order. We have studied the two following problems : where E ⊂ R2×2 is a compact isotropic set and where We show some existence theorems of W1,∞ and W2,∞ solutions respectively. Moreover we have studied a minimization problem : the computation of α(p, q, r) defined by

EPFL2004