ODE - Numerical Analysis

from pylab import *

Given an initial condition $y_0 \in \re^d$ and a right hand side function

f : (t,y) \to f(t,y) \in \re^d

consider the initial value problem

\begin{align*} y' &= f(t,y), \quad t \ge t_0 \\ y(t_0) &= y_0 \end{align*}

$f$ is continuous in $t$ and $y$ and Lipschitz continuous in $y$

\| f(t,x) - f(t,y) \| \le \lambda \| x - y \|, \qquad x,y \in \re^d, \quad t \ge t_0

For any $y_0 \in \re^d$ , there exists a unique solution. This solution is differentiable.
The solution depends continuously on the initial data; if $\hat y$ also satisfies the ODE but with initial condition $\hat y_0$ , then

\|y(t) - \hat y(t) \| \le \ee^{\lambda(t-t_0)} \| y_0 - \hat y_0 \|

\hat y' = f(t,\hat y) + r(t,\hat y)

where $r$ is bounded

\| r \| \le M

then

\|y(t) - \hat y(t) \| \le \ee^{\lambda(t-t_0)} \| y_0 - \hat y_0 \| + \frac{M}{\lambda} \left( \ee^{\lambda(t-t_0)} - 1 \right)

10 ODE