Picard Iteration and Successive Approximations

Picard iteration is a constructive method that simultaneously proves existence, uniqueness, and provides a practical algorithm for approximating solutions to initial value problems.

The Picard Iteration Scheme

Definition8.3Picard iterates

Given the IVP $\mathbf{x}' = \mathbf{F}(t, \mathbf{x})$ , $\mathbf{x}(t_0) = \mathbf{x}_0$ , the Picard iterates are defined recursively by:

$\mathbf{x}_0(t) = \mathbf{x}_0, \quad \mathbf{x}_{n+1}(t) = \mathbf{x}_0 + \int_{t_0}^{t} \mathbf{F}(s, \mathbf{x}_n(s))\,ds.$

This defines a sequence of continuous functions $\{\mathbf{x}_n\}_{n=0}^\infty$ on an interval around $t_0$ .

ExamplePicard iteration for a linear equation

For $x' = x$ , $x(0) = 1$ :

$x_0(t) = 1, \quad x_1(t) = 1 + \int_0^t 1\,ds = 1 + t,$ $x_2(t) = 1 + \int_0^t (1+s)\,ds = 1 + t + \frac{t^2}{2},$ $x_n(t) = \sum_{k=0}^{n} \frac{t^k}{k!} \to e^t \text{ as } n \to \infty.$

The Picard iterates converge to $e^t$ , which is the exact solution.

Convergence Analysis

Theorem8.2Convergence of Picard iterates

Under the hypotheses of the Picard-Lindelof theorem (with Lipschitz constant $L$ and bound $M$ ), the Picard iterates satisfy:

$\|\mathbf{x}_{n+1}(t) - \mathbf{x}_n(t)\| \leq \frac{ML^n|t - t_0|^{n+1}}{(n+1)!}$

for $|t - t_0| \leq \delta$ . Consequently, the series $\mathbf{x}_0 + \sum_{n=0}^{\infty}(\mathbf{x}_{n+1} - \mathbf{x}_n)$ converges uniformly and absolutely to a continuous function $\mathbf{x}(t)$ that solves the IVP.

RemarkRate of convergence

The bound $ML^n|t-t_0|^{n+1}/(n+1)!$ shows that the convergence is at least as fast as the series for $Me^{L|t-t_0|}$ . For practical computation, the number of correct digits roughly doubles with each iteration when $L|t-t_0|$ is moderate. The convergence rate improves on shorter intervals and deteriorates for stiff systems (large $L$ ).

The Contraction Mapping Perspective

Definition8.4Contraction mapping

Let $(X, d)$ be a complete metric space. A mapping $T: X \to X$ is a contraction if there exists $0 \leq q < 1$ such that $d(T(x), T(y)) \leq q \cdot d(x, y)$ for all $x, y \in X$ . By the Banach fixed-point theorem, $T$ has a unique fixed point $x^* = T(x^*)$ , and the iterates $T^n(x_0) \to x^*$ for any starting point $x_0$ .

RemarkPicard iteration as contraction mapping

Define the Picard operator $T$ on $C([t_0 - \delta, t_0 + \delta]; \mathbb{R}^n)$ by:

$T[\mathbf{x}](t) = \mathbf{x}_0 + \int_{t_0}^{t} \mathbf{F}(s, \mathbf{x}(s))\,ds.$

With the supremum norm and $\delta$ chosen so that $L\delta < 1$ , we have:

$\|T[\mathbf{x}] - T[\mathbf{y}]\|_\infty \leq L\delta \|\mathbf{x} - \mathbf{y}\|_\infty.$

So $T$ is a contraction with $q = L\delta < 1$ . Its unique fixed point is the solution to the IVP. This abstract viewpoint gives existence, uniqueness, and convergence of iterates in one stroke.

ExamplePicard iteration for a nonlinear ODE

For $x' = 1 + x^2$ , $x(0) = 0$ (whose exact solution is $x(t) = \tan t$ ):

$x_0 = 0, \quad x_1 = t, \quad x_2 = t + \frac{t^3}{3}, \quad x_3 = t + \frac{t^3}{3} + \frac{2t^5}{15} + \frac{t^7}{63}.$

These are the first terms of the Taylor series for $\tan t$ , illustrating how Picard iteration recovers the Taylor expansion term by term.