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

displaymath35

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

displaymath37

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:

(1)
tex2html_wrap_inline41 for every x;
(2)
then the recurrent formula holds

displaymath33

for tex2html_wrap_inline45 .

Example: Find the approximated sequence tex2html_wrap_inline39, for the IVP

displaymath49.

Solution: First let us write the associated integral equation

displaymath51

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

displaymath57

We have tex2html_wrap_inline59 , and

displaymath61

We leave it to the reader to show that

displaymath63

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

displaymath65

[Differential Equations] [First Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour