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## MATH-M343/S343 Section 2.8 Notes

by: Kathryn Brinser

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0

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# MATH-M343/S343 Section 2.8 Notes MATH-S343

Marketplace > Indiana University > Mathematics > MATH-S343 > MATH M343 S343 Section 2 8 Notes
Kathryn Brinser
IU
GPA 4.0

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Covers the method of Picard iterations for approximating.
COURSE
Honors Differential Equations
PROF.
Michael Jolly
TYPE
Class Notes
PAGES
1
WORDS
CONCEPTS
math-s343, math-m343, Picard, Approximation
KARMA
25 ?

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This 1 page Class Notes was uploaded by Kathryn Brinser on Wednesday September 21, 2016. The Class Notes belongs to MATH-S343 at Indiana University taught by Michael Jolly in Fall 2016. Since its upload, it has received 3 views. For similar materials see Honors Differential Equations in Mathematics at Indiana University.

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Date Created: 09/21/16
S343 Section 2.8 Notes- The Method of Successive Approximation (Picard Iterations) 9-15-16 ???????? = ????(????,???? ???? )  Applies to {???? ???? 0 = 0  Call solution ???? = ???? ???? o ???? ???? = ∫ ???? ???????????????? 0 ???????? = ∫ ????????(????,???? ???? )???????? 0  Start with initial approximatio0 ???? ???? = 0 for all ???? ???? o ????1???? =) ∫0 ????(????,???? 0 )???????? ???? o ???? ????+1 = ∫0????(????,???? ???? )???????? o Approximations become better as ???? → ∞  Ex. Approximate the first 3 Picard iterations of ???? ????,???? = ???? = 2???? − 1, ???? 0 = 1. o Must shift to satisfy correct form of initial condition  Let ???? = ???? − 1  ???? = 2 ???? + 1 − 1 substituted ???? = ???? + 1 into 2???? − 1 = 2???? + 1  ???? 0 = 1 − 1 = 0 ′  Apply iterations to {= 2???? + 1 ???? 0 = 0 ???? o ????1???? =) ∫0 2 0 + 1???????? ???? = ∫0 1???????? = ???? ???? o ???? 2 =) ∫0 2???? + 1???????? 2 ???? = ???? + ???? |0 = ???? + ???? ???? 2 o ???? 3 =) ∫0 2 ???? + ???? + 1???????? ???? 2 = ∫0 2???? + 2???? + 1???????? 2 ???? = ???? + ???? + ????| 3 0 = ???? + ???? + ???? 3 o Recall from 2.7 that solution is ???? = ????+ 1 2 2 3 o Recall that Taylor series for ???? = 1 + ???? + + ???? + ⋯ 2! 3! 2???? (2????) 2????)3  ???? = 1 + 2???? + 2! + 3! + ⋯ 1 (2????) (2????) 1  Exact solution here for ???? = ???? − 1 is ???? ???? =(1 + 2???? + + + ⋯ ) − 1 2 3 2 2! 3! 2 = 1 (1 + 2???? + 4???? + 8???? + ⋯ −) 1 2 2 6 2 = (1 + ???? + ???? + ???? + ⋯ − ) 1 2 3 2 = ???? + ???? + ???? + ⋯ 3 ????????  Theorem- If ????,???????? continuous in ???? = ????,???? × ????,???? centered about origin in ???????? plane, then the IVP ???????? ???????? = ????(????,???? ???? ) { has a unique solution for ???? in −ℎ,ℎ for some ℎ > 0 ???? 0 = 0 o Must have at least Lipschitz continuity- keep ???? fixed and take 2 different values of ????:  |???? ????,????1)− ???? ????,???? 2)|≤ ???? ???? 1 ???? 2| where constant ???? works no matter where ????1,???? 2re in ????  If????????continuous in ????, then | | ≤ ???? for some ???? ???????? ???????? ????????  By mean value theorem: ???? ????,???? 1)− ???? ????,???? 2)|= ???? ????????(???? 1 ???? 2)

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