Numerical stability in solution of ordinary differential equations

Matlab : Numerical Solution of Ordinary Differential Equations

Matlab has facilities for the numerical solution of ordinary differential equations (ODEs) of any order. In this document we first consider the solution of a first order ODE. Higher order ODEs can be solved using the same methods, with the higher order equations first having to be reformulated as a system of first order equations. Techniques for solving the first order and second order equation...

Numerical solution of ordinary differential equations

Ordinary differential equations are ubiquitous in science and engineering: in geometry and mechanics from the first examples onwards (Newton, Leibniz, Euler, Lagrange), in chemical reaction kinetics, molecular dynamics, electronic circuits, population dynamics, and many more application areas. They also arise, after semidiscretization in space, in the numerical treatment of time-dependent parti...

Numerical Solution of Ordinary Differential Equations

The Accurate Numerical Solution of Highly Oscillatory Ordinary Differential Equations*

An asymptotic theory for weakly nonlinear, highly oscillatory systems of ordinary differential equations leads to methods which are suitable for accurate computation with large time steps. The theory is developed for systems of the form Z = (A(t)/e)Z + H(Z,t). Z(0, f) = Z„, 0</< 7\0<e« 1, where the diagonal matrix A(t) has smooth, purely imaginary eigenvalues and the components of H(Z, i) are p...

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 f...