Numerical Analysis of Ordinary Differential Equations

Since many ordinary differential equations (ODEs) do not have a closed solution, approximating them is an important problem in numerical analysis. This work formalizes a method to approximate solutions of ODEs in Isabelle/HOL. We formalize initial value problems (IVPs) of ODEs and prove the existence of a unique solution, i.e. the Picard-Lindelöf theorem. We introduce general one-step methods for numerical approximation of the solution and provide an analysis regarding the local and global error of one-step methods. We give an executable specification of the Euler method to approximate the solution of IVPs. With user-supplied proofs for bounds of the differential equation we can prove an explicit bound for the global error. We use arbitrary-precision floating-point numbers and also handle rounding errors when we truncate the numbers for efficiency reasons. 1 Relations to the paper Our paper [1] is structured roughly according to the sources you find here. In the following list we show which notions of the paper correspond to which parts of the source code: • Arbitrary Precision Floats (Float): Included in the Library of the Isabelle/HOL distribution and in section ‘Floating-Point Numbers’ • Bounded Continuous Functions (C): Typedef bcontfun in Section 4 • Initial Value Problems in Section 5.2: – IVP ivp: Definition as locale ivp – is-solution, solution, unique-solution: Definition is-solution, Definition solution, Locale unique-solution – Combining solutions in Section 5.2.1