Picard's Method for Ordinary Differential Equations
This Demonstration constructs an approximation to the solution to a first-order ordinary differential equation using Picard's method. You can choose the derivative function using the drop-down menu and the initial guess for the algorithm. Increasing the number of iterations displayed using the slider shows closer approximations to the true solution, colored blue in the plot. The mean squared error (MSE) at each iteration is shown on the right.
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 .