Ordinary Differential Equations

Title:	Ordinary Differential Equations
Authors:	Fabian Immler and Johannes Hölzl
Submission date:	2012-04-26
Abstract:	Session Ordinary-Differential-Equations formalizes ordinary differential equations (ODEs) and initial value problems. This work comprises proofs for local and global existence of unique solutions (Picard-Lindelöf theorem). Moreover, it contains a formalization of the (continuous or even differentiable) dependency of the flow on initial conditions as the flow of ODEs. Not in the generated document are the following sessions: HOL-ODE-Numerics: Rigorous numerical algorithms for computing enclosures of solutions based on Runge-Kutta methods and affine arithmetic. Reachability analysis with splitting and reduction at hyperplanes. HOL-ODE-Examples: Applications of the numerical algorithms to concrete systems of ODEs. Lorenz_C0, Lorenz_C1: Verified algorithms for checking C1-information according to Tucker's proof, computation of C0-information.
Change history:	[2014-02-13]: added an implementation of the Euler method based on affine arithmetic [2016-04-14]: added flow and variational equation [2016-08-03]: numerical algorithms for reachability analysis (using second-order Runge-Kutta methods, splitting, and reduction) implemented using Lammich's framework for automatic refinement [2017-09-20]: added Poincare map and propagation of variational equation in reachability analysis, verified algorithms for C1-information and computations for C0-information of the Lorenz attractor.
BibTeX:	@article{Ordinary_Differential_Equations-AFP, author = {Fabian Immler and Johannes Hölzl}, title = {Ordinary Differential Equations}, journal = {Archive of Formal Proofs}, month = apr, year = 2012, note = {\url{http://isa-afp.org/entries/Ordinary_Differential_Equations.html}, Formal proof development}, ISSN = {2150-914x}, }
License:	BSD License
Depends on:	Affine_Arithmetic, List-Index, Triangle
Used by:	Differential_Dynamic_Logic, Hybrid_Systems_VCs
Status: [ok]	This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.

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