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

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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 differential 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 off error)? How many steps are required? (Just predict — no need to test it.) ( c ) Answer part ( b ) for the Improved Euler Method. ( d ) Answer part ( b ) for the fourth order Runge–Kutta Method. ( e ) Discuss the behavior of the solution, both analytical and numerical, for the alternative initial condition u (0) = - . 1.

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

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