Picard iteration

Summary

Picard iteration is the construction of a solution of $x' = f(t, x)$, $x(t_0) = x_0$ as a limit of successive approximations. One starts with the constant curve $\varphi_0(t) = x_0$ and sets $\varphi_{k+1} = P \varphi_k$, where $P$ is the Picard operator: $\varphi_{k+1}(t) = x_0 + \int_{t_0}^{t} f(s, \varphi_k(s))\, ds$. Under the Picard–Lindelöf hypotheses, the curves $\varphi_k$ converge uniformly on the time interval to the unique solution.

Remarks

Example: for $x' = x$, $x(0) = 1$, the iterates are $\varphi_k(t) = \sum_{j=0}^{k} t^j / j!$. They converge to $e^t$. Let $K$ be the Lipschitz constant and $T$ the length of the time interval. The error after $k$ steps is at most $C (K T)^{k} / k!$ for a constant $C$. See an iterate of the Picard operator is a contraction.

Explanations

L1 human

Start with a rough guess, the constant $x_0$. Put the guess into the right side of the integral equation to get a better guess, and repeat. Each step improves the approximation, and the guesses converge to the solution.