Picard's Iteration Method

Picard iteration turns an initial value problem into a sequence of integral approximations. It is important less as a production ODE solver than as the constructive idea behind a fundamental existence-and-uniqueness theorem.

Consider

$$ y'(t) = f(t,y(t)), \qquad y(t_0) = y_0. $$

Integrating from $t_0$ to $t$ gives the equivalent integral equation

$$ y(t) = y_0 + \int_{t_0}^{t} f(s,y(s))\, ds. $$

Picard's method replaces the unknown function inside the integral with the previous approximation.

Picard Iteration

Choose an initial function, often

$$ y_0(t) = y_0. $$

Then define

$$ y_{n+1}(t) = y_0 + \int_{t_0}^{t} f(s,y_n(s))\, ds. $$

Each iteration produces a new function, not merely a single number.

Fixed-Point Interpretation

Define an operator

$$ (Ty)(t) = y_0 + \int_{t_0}^{t} f(s,y(s))\, ds. $$

A solution of the IVP is exactly a fixed point:

$$ Ty = y. $$

Picard iteration is therefore the fixed-point iteration

$$ y_{n+1} = Ty_n. $$

If $T$ is a contraction on a suitable function space, the contraction mapping theorem guarantees convergence to a unique fixed point.

Why the Lipschitz Condition Appears

Suppose

$$ |f(t,y) - f(t,z)| \le L|y - z|. $$

Then on a short interval $|t-t_0|\le a$,

$$ |(Ty)(t) - (Tz)(t)| \le \int_{t_0}^{t} L|y(s) - z(s)|\, ds. $$

Using the supremum norm,

$$ \| Ty - Tz\|_\infty \le La\| y - z\|_\infty. $$

If

$$ La<1, $$

then $T$ is a contraction. This is the core idea behind the local Picard--Lindelof theorem.

Worked Example

Consider

$$ y' = x + y, \qquad y(0) = 1. $$

The integral form is

$$ y(x) = 1 + \int_0^x(t + y(t))\, dt. $$

Start with

$$ y_0(x) = 1. $$

The first iterate is

$$ y_1(x) = 1 + \int_0^x(t + 1)\, dt = 1 + x + \frac{x^2}{2}. $$

The second iterate is

$$ y_2(x) = 1 + \int_0^x \left(t + 1 + t + \frac{t^2}{2} \right)dt, $$

so

$$ y_2(x) = 1 + x + x^2 + \frac{x^3}{6}. $$

The third iterate is

$$ y_3(x) = 1 + x + x^2 + \frac{x^3}{3} + \frac{x^4}{24}. $$

The exact solution of

$$ y' - y = x $$

is

$$ y(x) = 2e^x - x - 1. $$

The iterates approach this solution on intervals where the contraction argument applies.

An Important Distinction

Picard iteration is sometimes described as a numerical method, but a direct implementation can be expensive because every iteration requires a new integral over an entire function.

In practical numerical integration, methods such as Runge--Kutta usually provide better efficiency. Picard iteration remains especially valuable for:

• proving local existence and uniqueness,
• deriving series approximations,
• understanding fixed-point methods,
• motivating iterative schemes for integral and differential equations.

Numerical Implementation

If the integral cannot be evaluated analytically, choose grid points

$$ t_0<t_1<\cdots<t_m $$

and approximate

$$ \int_{t_0}^{t_j}f(s,y_n(s))\, ds $$

with a quadrature rule.

This creates two sources of error:

1. iteration error from stopping after finitely many Picard steps,
2. quadrature error from approximating the integral.

The two errors should not be confused.

Convergence Caveats

Picard iteration may fail or become inconvenient when:

• $f$ is not Lipschitz in the relevant region,
• the interval is too large for the contraction estimate,
• iterates leave the region where the assumptions hold,
• evaluating the required integrals is difficult,
• the ODE is stiff or highly nonlinear.

A local convergence theorem does not imply that one Picard iteration run converges uniformly over an arbitrarily long interval.