aimz6 - AIMS Exercise Set # 6 Peter J. Olver 1. Prove that...

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AIMS Exercise Set # 6 Peter J. Olver 1. Prove that the Midpoint Method (10.58) is a second order method. 2. Consider the initial value problem du dt = u (1 - u ) , u (0) = . 1 , for the logistic diferential equation . ( a ) Find an explicit formula for the solution. Describe in words the behavior of the solution for t > 0. ( b ) Use the Euler Method with step sizes h = . 2 and . 1 to numerically approximate the solution on the interval [0 , 10]. Does your numerical solution behave as predicted from part ( a )? What is the maximal error on this interval? Can you predict the error when h = . 05? Test your prediction by running the method and computing the error. Estimate the step size needed to compute the solution accurately to 10 decimal places (assuming no round o± error)? How many steps are required? (Just predict — no need to test it.) ( c ) Answer part ( b ) for the Improved Euler Method. ( d
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This note was uploaded on 02/10/2012 for the course MATH 5485 taught by Professor Olver during the Fall '09 term at University of Central Florida.

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aimz6 - AIMS Exercise Set # 6 Peter J. Olver 1. Prove that...

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