DE.04.4.Literacy

# DE.04.4.Literacy - 48 iterations Differential...

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Mathematica Authors: Bruce Carpenter, Bill Davis and Jerry Uhl ©2001-2007 Publisher: Math Everywhere, Inc. Version 6.0 DE.04 Modern DiffEq Issues LITERACY · L.1) When you start a car trip, you go to where the car is parked, start the car, and drive off. Your trip is completely determined by your steering wheel, gas pedal and brake pedal actions during your trip. Comment on this: When you are given function f @ t, y D and make a diffeq y £ @ t D = f @ t, y @ t DD with y @ 0 D given, then the plot of the corresponding solution is determined in much the same way that a car trip is determined. · L.2) Here's an attempt to use Euler's faker (= Euler's method) to approximate f @ x D = 1 2 x Sin @ 3 x D using only information about f £ @ x D and the value of f @ 0 D . 1 2 3 4 x - 2.0 - 1.5 - 1.0 - 0.5 0.5 1.0 6 iterations What you see above is what you get when you go with 6 iterations. Here's what you get when you go with 24 iterations: 1 2 3 4 x - 1.5 - 1.0 - 0.5 0.5 1.0 24 iterations 48 iterations: 1 2 3 4 x - 1.5 - 1.0 - 0.5 0.5 1.0 48 iterations What is it about Euler's method (= Euler's faker) that makes it so that you can always expect to get better results when you increase the number of iterations? · L.3) You are looking at the diffeq y £ @ t D = y @ t D - t with y @ 0 D = 1. When you go with Euler's method starting at 8 0, 1 < and a jump size = 1, then the first three Euler points you get are 8 0, 1 < , 8 1, a < and 8 2, b < . Give the precise values of a and b.

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## This note was uploaded on 09/21/2011 for the course MATH 285 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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DE.04.4.Literacy - 48 iterations Differential...

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