14. A little bit of theory

An ordinary differential equation (ODE) is given by

\[ y^\prime (t) = f(t,y(t)) \qquad \forall \, t \in (t_0, T) \]

together with the initial condition

\[ y(t_0) = y_0, \]

where

• \(f : [t_0, T] \times {\mathbb R}^n \rightarrow {\mathbb R}^n\) is the given right hand side

• \(y : [t_0, T] \rightarrow {\mathbb R}^n\) is the unknown function, called state

• \(t \in [t_0, T]\) is called time

• \(y_0 \in {\mathbb R}^n\) is the initial value

14.1. Some examples:

• \(n = 1\), \(f(t,y) = a y\):

\[ y^\prime(t) = a y, \quad y(t_0) = y_0 \]

This ode has the solution \(y(t) = y_0 e^{a (t-t_0)}\). If \(a > 0\), the solution is exponentiall increasing, for \(a < 0\), it is exponentially decreasing.

• Consider \(y^{\prime \prime}(t) = -\omega^2 y(t)\) with \(y(0) = y_0\), \(y^\prime(0) = y_{0p}\). This is a second order ODE, which can be reduced to a first oder system with the definition \(y_1 := y\) and \(y_2 := y^\prime/\omega\):

\[\begin{split} \left( \begin{array}{c} y_1 \\ y_2 \end{array} \right)^\prime = \left( \begin{array}{c} \omega y_2 \\ -\omega y_1 \end{array} \right) \quad \text{with i.c.} \quad \left( \begin{array}{c} y_1 \\ y_2 \end{array} \right)(0) = \left( \begin{array}{c} y_0 \\ y_{0p}/\omega \end{array} \right) \end{split}\]

Its solution is \(y(t) = y_0 \cos( \omega t ) + \frac{y_{0p}}{\omega} \sin (\omega t)\).

14.2. Some Theorems

Theorem: Picard Lindelöf

Assume the right hand side satisfies a Liptschitz condition in the second argument

\[ \| f(t, y) - f(t, z) \| \leq L \, \| y - z \| \qquad \forall \, t \in [t_0, T], \, \forall \, y,z \in {\mathbb R}^n. \]

Then there exisits a unique solution of the ODE.

Theorem: Stability with respect to initial conditions

Consider two ODEs differing only in the initial values:

\[\begin{split} y^\prime(t) = f(t, y(t)), \quad y(t_0) = y_0 \\ \end{split}\]

\[\begin{split} z^\prime(t) = f(t, z(t)), & & \quad z(t_0) = z_0 \\ \end{split}\]

Then there holds:

\[ \| y(t) - z(t) \| \leq e^{L (t-t_0)} \| y_0 - z_0 \| \]

This means that an error in the initial condition may be propagated with exponential groth with constant \(L\).

Theorem: One-sided Lipschitz condition

Assume there holds

\[ \left< f(t,y)-f(t,z), y-z \right> \leq \alpha \| y - z \|_2^2 \]

with some \(\alpha \in {\mathbb R}\). Then the stability estimate can be improved to

\[ \| y(t) - z(t) \|_2 \leq e^{\alpha (t-t_0)} \| y_0 - z_0 \|_2 \]

This is an essential improvement for the first example above when \(a < 0\). The second example satisfies the one-sided Lipschitz condition with \(\alpha = 0\).

14.3. Autonomous ODEs

An ODE is called autonomous iff the right hand side does not depend explicitely on \(t\), i.e. \(y^\prime(t) = f(y(t))\). Then the time axis can be shifted arbitrarily.

A non-autonomous ODE in \({\mathbb R}^n\) can be transformed to an autonomous one in \({\mathbb R}^{n+1}\) by letting \(y_{n+1}(t) := t\) be another unknown state variable, which satisfies the ODE \(y_{n+1}^\prime = 1\) and i.c. \(y_{n+1}(t_0) = t_0\).

14.4. An equivalent integral equation

A smooth solution of the ODE satisfies the following integral equation (IE), and vice versa:

\[ y(t) = y_0 + \int_{t_0}^t f(s, y(s)) ds \qquad \forall \, t \in [t_0, T] \]

We call the ODE and the IE formally equivalent. The IE may have a solution which is not differentiable, and thus not a solution of the ODE (e.g. if \(f\) jumps in some \(\overline t\), then \(y\) has a kink, and is not differentiable).

The integral equation is the basis for analysis as well as designing numerical methods.