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project1 - ECE633 Project 1 Due 1 Given a scalar...

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Pekarek-ECE633 1/24/12 1 ECE633 - Project 1, Due January 31, 2012 1) Given a scalar differential equation (i.e. ) , For a p -th order one-step method, a function that satisfies and is called the principal error function . Derive for the Forward Euler algorithm. (Hint: use the Taylor series approach shown on pages 4b and 5 of Lecture 2). 2) Given a scalar differential equation (i.e. ) , (real ). a) Show that as for any initial . b) Prove that this solution is unique. c) Under what conditions on can one assert that and for a Runge-Kutta 4th order solution? d) What is the analogous result for a Forward Euler Algorithm? e) Generalize these results to where is a constant matrix all of whose eigenvalues have negative real parts. 3) Consider the initial value problem: , . a) Determine , , and analytically. b) Write a MATLAB routine to perform a fixed-step Forward Euler and fixed-step 4th-order Runge Kutta Algorithm for this problem. For both algorithms, solve the problem from with a fixed step length h = 1/200. Plot the analytical solution of a) with those

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project1 - ECE633 Project 1 Due 1 Given a scalar...

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