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