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:
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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Author: Mohamed Amine Khamsi