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Wednesday, 27 March 2013

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
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It is not hard to see that the solution to this problem is also given as a solution to (called the integral associated equation)
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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 holdsdisplaymath33
for tex2html_wrap_inline45 .
Example: Find the approximated sequence tex2html_wrap_inline39, for the IVP
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Solution: First let us write the associated integral equation
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Set tex2html_wrap_inline53 . Then for any tex2html_wrap_inline45 , we have the recurrent formula
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We have tex2html_wrap_inline59 , and
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We leave it to the reader to show that
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