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Course: MATH 612, Fall 2008
School: Maryland
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Word Count: 203

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612 AMSC Numerical Methods for Partial Dierential Equations Spring Term 2004 Instructor: Georg Dolzmann Homework set #1 Problem 1: [Morton&amp;Meyers 2.1] (i) The function u0 (x) is dened on [0, 1] by u0 (x) = 2x 2 2x 1 if 0 x 2 , 1 if 2 x 1. Show that the Fourier sine series of u0 is given by u0 (x) = m=1 am sin mx where am = (ii) Show that 2p+2 2p m 8 sin . m2 2 2 1 2 dx 2 x (2p + 1)2 1 1...

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612 AMSC Numerical Methods for Partial Dierential Equations Spring Term 2004 Instructor: Georg Dolzmann Homework set #1 Problem 1: [Morton&Meyers 2.1] (i) The function u0 (x) is dened on [0, 1] by u0 (x) = 2x 2 2x 1 if 0 x 2 , 1 if 2 x 1. Show that the Fourier sine series of u0 is given by u0 (x) = m=1 am sin mx where am = (ii) Show that 2p+2 2p m 8 sin . m2 2 2 1 2 dx 2 x (2p + 1)2 1 1 < . 2 (2p + 1) 4p0 and hence that p=p0 0 (iii) Deduce that u (x) is approximated on the interval [0, 1] to within 0.001 by the sine series in part (i) truncated after m = 405. Problem 2: Write a MATLAB program to solve the equation heat ut = uxx u(0, t) = 0 u(1, t) = 0 0 in (0, 1) (0, ), for t > 0, for t > 0, u(x, 0) = u (x) for x (0, 1), where u0 (x) = 2x 2 2x 1 if 0 x 2 , 1 if 2 x 1. Implement the explicit scheme that leads to n+1 n n n n Uj = Uj + Uj+1 2Uj + Uj1 2 for j = 1, . . . , J 1, and n 1. Here = t/(x)2 and 0 Uj = u0 (xj ), j = 1, 2, . . . , J 1, n n U0 = UJ = 0...

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AMSC 612 Numerical Methods for Partial Dierential Equations Spring Term 2004 Instructor: Georg Dolzmann Homework set #3Problem 1: [Morton&amp;Meyers 2.7] Show that the leading order term in the truncation error of the explicit schemen+1 n n n n n Uj
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