Scientific Computing

This module illustrates Picard iteration for solving an initial
value problem for an ordinary differential equation. Picard iteration
generates a solution of an initial value problem for an ordinary
differential equation (ODE) using fixed-point iteration. Picard
iteration is used mainly as a theoretical tool to prove the existence
of a solution, but it can be implemented numerically, as demonstrated
in this module. Given an ODE *y*′ =
*f*(*t*, *y*)*y*(*t*_{0}) = *y*_{0}*integral equation*

which must be satisfied by the solution *y* of the initial value
problem. Solving the ODE thus appears to have become a quadrature
problem, except that we don't know the value of *y* on the
right-hand side. This integral equation formulation is set up
naturally for fixed-point iteration, however, plugging a guess for
*y* into the right-hand side to obtain a new value for *y*
and then repeating. Starting with the constant function
*y*_{0}(*t*) = *y*_{0}

If the function *f* satisfies a Lipschitz condition, then it
can be shown that there is an interval about
*t*_{0}*y* of the initial value problem. To implement
Picard iteration numerically, the required integrals can be computed
using a composite quadrature rule.

The user begins by selecting a differential equation from the menu
provided. The exact solution curve through a predefined initial value
is drawn in black, and the constant starting function for Picard
iteration (based on the prescribed initial value) is drawn in blue.
The user also selects a quadrature rule from the menu provided and a
value for the mesh spacing of the composite quadrature using the
slider. Each time the user clicks *Iterate*, a Picard iteration
is performed and the resulting new approximate solution is added to the
graph. Successive approximate solutions are color coded from blue to
red as iterations proceed. Note that the interval in which Picard
iteration converges to the true solution does not always extend across
the entire range shown.

**Reference:** R. M. M. Mattheij and J. Molenaar, *Ordinary
Differential Equations in Theory and Practice*, John Wiley & Sons,
New York, 1996 (reprinted by SIAM, 2002). See pages 30-31.

**Developers:** Evan VanderZee and Michael Heath