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Approximating the Solution of Ordinary Differential Equations Using Picard's Method

An approximation to the solution to a first order ordinary differential equation is constructed using Picard's iterative method. The derivative function may be chosen using the drop down menu, as well as the initial guess in the algorithm. Increasing the number of iterations shown using the slider demonstrates the approach of the approximation to the true solution, shown in blue in the plot. The mean square error at each iteration is shown on the right.

Contributed by: Oliver K. Ernst

SNAPSHOTS

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 succesive approximations to the solution as

such that after the iteration .

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 .