Picard Iteration

Introduction

    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  .  In differential equations, Picard iteration is a constructive procedure for establishing the existence of a solution to a differential equation   that passes through the point  .

    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."  For years,  see references [3 to 7].   

Background

    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    are continuous on the rectangle    and  ,  then there exists a unique solution to the initial value problem (I.V.P.)

(1)          

for all values of  x  in some (smaller) interval   contained in  .  

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    of functions

        ,   and
(2)    
        .

Then the solution to (1) is given by the limit:

(3)        .

Symbolic Analysis

    Computer algebra systems are useful for performing the arduous task of repeated integrations.  To perform Picard iteration using Mathematica one must define  ,  and supply the initial value  ,  and the number of iterations n.  The following subroutine will compute the first approximations.

Mathematica Subroutine (Picard Iteration).

Getting the Graphs

    If we want to graph several solution, we will need to store them somewhere.  The following version of the Picard subroutine uses the vector  .  Since the subscript for the elements of a vector start with , the notation for the sequence of stored functions will be  .  

Mathematica Subroutine (Vector Form for Picard Iteration).

Extension to First Order Systems in 2D

    Suppose that we want to solve the initial value problem for a system of two differential equations

        ,    and  
(7)
          

Picard iteration can be used to generate two sequences    and    which converge to the solutions    and  ,  respectively, see reference [2].  They are defined recursively by

        ,  
        ,   and
(8)
           

The sequence of approximations will converge to the solution, i.e.

(9)