Picard Iterative Process

Indeed, often it is very hard to solve differential equations, but we do have a numerical process that can approximate the solution. This process is known as the Picard iterative process.
First, consider the IVP

It is not hard to see that the solution to this problem is also given as a solution to (called the integral associated equation)

The Picard iterative process consists of constructing a sequence of functions which will get closer and closer to the desired solution. This is how the process works:

(1)

for every x;

(2)

then the recurrent formula holds

for .

Example: Find the approximated sequence , for the IVP

.

Solution: First let us write the associated integral equation

Set . Then for any , we have the recurrent formula

We have , and

We leave it to the reader to show that

We recognize the Taylor polynomials of (which also get closer and closer to) the function

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