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Unformatted text preview: MAA 4212, Spring 2009—Miscellaneous nonbook problems F1. Picard iteration. Recall that our proof of existence and uniqueness of solutions to the initial value problem dy dt = f ( t,y ) , y ( t ) = y (1) used the Contracting Mapping FixedPoint Theorem (CMT), which was itself proved by looking at the sequence of iterates of a point under a contraction defined on some complete metric space. In the FTODE, the metric space was a closed ball in C ( I ) for some closed interval I , and the contraction was the map H defined by H ( g )( t ) = y + Z t t f ( s,g ( s )) ds. Recall that the CMT gives us not just existence and uniqueness of a fixed point, but a way of constructing the fixed point: start with any point in the metric space, and follow the sequence of points obtained by repeatedly applying the contraction. Carrying out this procedure in the context of the FTODE is called Picard iteration . We start with a function g (usually the constant function t 7→ y ) defined on some neighborhood of t , define g 1 = H ( g ) ,g 2 = H ( g 1 ), etc. The proof of the FTODE (via the CMT) shows that if we take δ small enough, (1) will have a unique solution on ( t δ,t + δ ), and the sequence { g n } will converge uniformly to this solution on that interval. Thus, Picard iteration gives us a (not necessarily efficient) way to produce the solution of (1). Of course, for some f we can solve (1) explicitly, in closed form, rather than express the solution as the limit of some sequence; that’s what you did in the first few weeks of MAP 2302. For such ODEs, there is no point to doing Picard Iteration other than for fun, to see what happens, or as an exercise in learning. (Even for ODEs that we can’t solve by MAP 2302 methods, there are usually much more efficient ways of computing solutions than to use Picard Iteration.) While the proof of the FTODE requires us to choose δ sufficiently small, in practice when we do Picard Iteration we don’t worry about how small δ needed to be for that...
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 Spring '08
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 Trigraph, CMT, Picard iteration, FTODE

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