Ordinary Differential Equations


Title: Ordinary Differential Equations
Authors: Fabian Immler and Johannes Hölzl
Submission date: 2012-04-26

Session HOL-ODE 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-Refinement: Intermediate session combining \texttt{HOL-ODE} and Lammich's (automatic) refinement framework.
  • 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_Approximation, 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.
  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, Collections, Deriving, List-Index, Show, Triangle
Used by: Akra_Bazzi, Bertrands_Postulate, Differential_Dynamic_Logic
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.