Description
The Picard-Lindelöf iteration is another way of solving Ordinary differential equations. The idea is to define an operator which is a Kontraktion, meaning it has one fixed point. This fixed points solves the equation.
Definition
Let be open and continuous and with respect to locally lipschitz function. We study the ODE The solution can be calculated or approximated using the Picard iteration. We define the functional operator Notice that since this is just the integral of our ODE every fixed point is a solution. Picard showed that under the given condition given will be a Kontraktion, meaning every converges to exactly one under iteration.
Proof sketch: I want to quickly prove that is a contraction. Everything else will be ignored for now. We do a chain of simple inequalities:
&\leq \sup_{t\in [a, b]}\int_{t_{0}}^{t} \| f(s, g(s))- f(s,h(s))\|ds \\ &\leq \sup_{t\in [a, b]}\left(|t-t_{0}|\sup_{s\in[t_{0}, t]}\|f(s, g(s))-f(s, h(s))\|\right)\\ &\leq \sup_{t\in [a, b]}\left(|t-t_{0}| L\sup_{s\in[t_{0}, t]}\|g(s)-h(s)\|\right)\\ &\leq L(b-a)\|g-h\|_\infty \end{align}$$ Where $\|\cdot\|_{\infty}$ is the infinity norm on some iinterval $[a, b]$ containing $t_{0}$. If $[a, b]$ is now chosen small enough (such that $\Theta = L(b-a) < 1$) then the inequality $\|T(g)-T(h)\|_{\infty} \leq \Theta \|g-h\|_\infty$ holds and we have a contraction. # Properties >[!tip] > > # Application >[!example] Application > >Start with a ansatz $x_{0}(t) = x_{0}$. Define the next functions as $x_{1}(t) = (Tx_{0})(t)$. Determine the limit $x_\infty$ of the sequence (which exists by the Picard).