Picard iteration

Picard iteration is a constructive method for finding the solution to an Initial value problem $y^{\prime }=f(x,y)$, $y(x_0)=y_0$. It’s how the Existence and uniqueness theorem is proved — by actually building the solution as the limit of a sequence of approximations.

The idea

Start with the IVP and convert it to an integral equation by integrating both sides from $x_0$ to $x$:

$y(x)=y_0+{\int }_{x_0}^{x}f(t,y(t))dt$

This gives the same information as the IVP but in integral form. The unknown $y$ appears on both sides — that’s why we iterate.

The Picard sequence:

• ${\varphi }_0(x)=y_0$ (constant).
• ${\varphi }_1(x)=y_0+{\int }_{x_0}^{x}f(t,{\varphi }_0(t))dt$.
• ${\varphi }_2(x)=y_0+{\int }_{x_0}^{x}f(t,{\varphi }_1(t))dt$.
• $⋮$
• ${\varphi }_{n+1}(x)=y_0+{\int }_{x_0}^{x}f(t,{\varphi }_n(t))dt$.

Each ${\varphi }_{n+1}$ is built by plugging the previous approximation into the integrand.

If the conditions of the Existence and uniqueness theorem are satisfied, the sequence ${\varphi }_n$ converges to the unique solution $\varphi$ of the IVP.

When you’ve reached the solution

If at some step ${\varphi }_{k+1}(x)={\varphi }_k(x)$ exactly, the iteration has converged — that’s the solution. Formally: if there exists $k\in \mathbb{N}$ such that ${\varphi }_{k+1}(x)={\varphi }_k(x)$, then

${\varphi }_k(x)=y_0+{\int }_{x_0}^{x}f(s,{\varphi }_k(s))ds$

is the solution of the IVP.

In practice, this only happens for very simple $f$. Usually you compute several iterations and identify a pattern (a Taylor series, an exponential, etc.) that the iterates are converging toward.

Worked example

The IVP $y^{\prime }=2t(1+y)$, $y(0)=0$.

Step 0: ${\varphi }_0(t)=0$.

Step 1: ${\varphi }_1(t)={\int }_0^{t}2s(1+0)ds=t^2$.

Step 2: ${\varphi }_2(t)={\int }_0^{t}2s(1+s^2)ds=t^2+\frac{t^4}{2}$.

Step 3: ${\varphi }_3(t)={\int }_0^{t}2s\left(1+s^2+\frac{s^4}{2}\right)ds=t^2+\frac{t^4}{2}+\frac{t^6}{2\cdot 3}$.

Pattern emerging: ${\varphi }_n(t)=t^2+\frac{t^4}{2!}+\frac{t^6}{3!}+\cdots +\frac{t^{2n}}{n!}={\sum }_{k=1}^n\frac{t^{2k}}{k!}$.

This is the partial sum of $e^{t^2}-1$. Confirming with the ratio test:

$\left|\frac{t^{2(k+1)}/(k+1)!}{t^{2k}/k!}\right|=\frac{t^2}{k+1}\to 0\text{as }k\to \infty$

So the series converges for every $t$, and $\varphi (t)={\sum }_{k=1}^{\infty }\frac{t^{2k}}{k!}=e^{t^2}-1$.

You can verify directly: ${\varphi }^{\prime }(t)=2t{e}^{t^2}=2t(1+(e^{t^2}-1))=2t(1+\varphi )$. ✓ And $\varphi (0)=0$. ✓

Why this works

Picard iteration is a fixed-point iteration on the operator $T\varphi =y_0+{\int }_{x_0}^{x}f(t,\varphi (t))dt$. The fixed point of $T$ — the function with $T\varphi =\varphi$ — satisfies the integral equation, hence the IVP.

The conditions in the Existence and uniqueness theorem don’t directly make $T$ a contraction — what they give you is Lipschitz continuity of $f$ in $y$ (continuity of $\partial f/\partial y$ on a closed rectangle implies a Lipschitz bound there). Lipschitz alone is not enough for $T$ to be a contraction in the sup norm; you also need the interval to be short enough. Specifically, if $L$ is the Lipschitz constant of $f$ in $y$, then $T$ is a contraction on functions defined on $[x_0-\delta ,x_0+\delta ]$ provided $L\delta <1$. Once you’ve shrunk to a short enough interval, Banach’s fixed-point theorem applies and gives convergence to a unique fixed point.

This is why the existence-and-uniqueness conclusion is local — the contraction only works on a small enough interval, set by the Lipschitz constant.

Practical relevance

Picard iteration is rarely used to compute solutions — for most ODEs, direct methods (Separable equation, Integrating factor, etc.) or numerical methods (Euler, Runge-Kutta) are vastly faster. Its importance is theoretical: it provides the existence-and-uniqueness proof, and it shows that solutions can be approximated systematically when nothing else works.

For numerical ODE-solving in code, see runge-kutta-style methods (not in this vault yet) — they’re spiritual cousins of Picard iteration but with much better convergence rates per step.