Tuesday, February 2, 2010

Picard Iteration

The term "Picard iteration" occurs two places in undergraduate mathematics. In numerical analysis it is used when discussing fixed point iteration for finding a numerical approximation to the equation [Graphics:Images/PicardIterationMod_gr_1.gif]. In differential equations, Picard iteration is a constructive procedure for establishing the existence of a solution to a differential equation [Graphics:Images/PicardIterationMod_gr_2.gif] that passes through the point [Graphics:Images/PicardIterationMod_gr_3.gif].

The first type of Picard iteration uses computations to generate a "sequence of numbers" which converges to a solution. We will not present this application, but mention that it involves the traditional role of computer software as a "number cruncher."

The goal of this article is to illustrate the second application of Picard iteration; i. e. how to use a computer to efficiently generate a "sequence of functions" which converges to a solution. We will see that computer software can perform the more sophisticated task of "symbol cruncher

Most differential equations texts give a proof for the existence and uniqueness of the solution to a first order differential equation. Then exercises are given for performing the laborious details involved in the method of successive approximations. The concept seems straightforward, just repeated integration, but students get bogged down with the details. Now computers can do all the drudgery and we can get a better grasp on how the process works.

Theorem 1 (Existence Theorem):
If both [Graphics:Images/PicardIterationMod_gr_4.gif] are continuous on the rectangle [Graphics:Images/PicardIterationMod_gr_5.gif] and [Graphics:Images/PicardIterationMod_gr_6.gif], then there exists a unique solution to the initial value problem (I.V.P.)

(1) [Graphics:Images/PicardIterationMod_gr_7.gif]

for all values of x in some (smaller) interval [Graphics:Images/PicardIterationMod_gr_8.gif] contained in [Graphics:Images/PicardIterationMod_gr_9.gif].

Picard's Method for D.E.'s

The method of successive approximations uses the equivalent integral equation for (1) and an iterative method for constructing approximations to the solution. This is a traditional way to prove (1) and appears in most all differential equations textbooks. It is attributed to the French mathematician Charles Emile Picard (1856-1941).

Theorem 2 (Successive Approximations - Picard Iteration):

The solution to the I.V.P in (1) is found by constructing recursively a sequence [Graphics:Images/PicardIterationMod_gr_10.gif] of functions

[Graphics:Images/PicardIterationMod_gr_11.gif], and
(2)
[Graphics:Images/PicardIterationMod_gr_12.gif].

Then the solution [Graphics:Images/PicardIterationMod_gr_13.gif] to (1) is given by the limit:

(3) [Graphics:Images/PicardIterationMod_gr_14.gif].

No comments:

Post a Comment

Introductory Methods of Numerical Analysis