Details

Snapshot 1: the approximations approach the true solution with increasing iterations of Picard's method

Snapshot 2: the approximation after the first iteration already captures the behavior of the solution

Snapshot 3: although the initial guess is poor, the approximations rapidly improve

Picard's method approximates the solution to a first-order ordinary differential equation of the form

,

with initial condition . The solution is

.

Picard's method uses an initial guess to generate successive approximations to the solution as

such that after the iteration .

Above, we take , with and . Several choices for the initial guess and differential equation are possible. After each iteration, the mean squared error of the approximation is computed by sampling the true solution (in blue) and the approximation at evenly spaced points in .