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 tex2html_wrap_inline39 of functions which will get closer and closer to the desired solution. This is how the process works:

tex2html_wrap_inline41 for every x;
then the recurrent formula holds


for tex2html_wrap_inline45 .

Example: Find the approximated sequence tex2html_wrap_inline39, for the IVP


Solution: First let us write the associated integral equation


Set tex2html_wrap_inline53 . Then for any tex2html_wrap_inline45 , we have the recurrent formula


We have tex2html_wrap_inline59 , 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

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