Picard operator

Definition

Let $J = [t_{\min}, t_{\max}]$, let $t_0 \in J$ and let $x_0 \in \mathbb{R}^n$. Let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be a function such that $s \mapsto f(s, \gamma(s))$ is integrable on $J$ for each continuous $\gamma : J \to \mathbb{R}^n$. The Picard operator of $f$ at $(t_0, x_0)$ maps a continuous curve $\gamma : J \to \mathbb{R}^n$ to the curve $P\gamma : J \to \mathbb{R}^n$ given by $$(P \gamma)(t) = x_0 + \int_{t_0}^{t} f\left(s, \gamma(s)\right) ds.$$

Remarks

Assume the Picard–Lindelöf hypotheses with constants $a, L, K$. Then $P$ maps the continuous curves with values in the closed ball $\|x - x_0\| \le a$ to curves in this ball. A fixed point of $P$ is a solution of the initial value problem. See the integral equation form. Picard iteration applies $P$ again and again, starting from the constant curve $x_0$.

Explanations

L1 human

The operator takes a guess for the solution and makes a new guess. It integrates the field along the old guess, starting from $x_0$. A curve that the operator does not change is a solution.