Friday, Sept. 25: Our first problem (taken from Chapter 2, Problem 2 in Farlow), was concerned with heat flow in a bar that could be decomposed into a steady state part and a time dependent part. The time dependent part has a Fourier series representation, and the Fourier coefficients are determined by the initial conditions. In the statement of the problem in Farlow, it was casually suggested that the initial condition might be temperature ZERO everywhere, and from this starting point the bar warms up to its steady state. Try finding the Fourier coefficients and actually computing the temperature distribution at later times.
Friday, Oct. 2: Consider the function f(x)=(x-1/2)2 on the interval [0,1]. Its graph is a parabola centered in the interval. Represent it as a Fourier sine series, using basis functions sin(m&pi x), and as a Fourier cosine series, using basis functions cos(m&pi x). Do this in more than one way: (1) by exact integration of the formula for Fourier coefficients, using Maple, (2) by numerical integration of the formula for Fourier coefficients, (3) by the FFT, using appropriate extensions outside the interval [0,1]. Be clear about how you are extending f. Note that (3) depends on N, the number of Fourier terms (which is also the number of points in the discretization of f), and (1) does not depend on this, so you should expect a slight difference in the coefficients you obtain. (2) of course also depends on N, although it is meant to be an approximation to (1), unlike (3), which is an exact representation of the digitized f. Try to work precisely, and notice and comment on little discrepancies, whether they are real differences in method or simply the result of roundoff error in the computation.Friday, Oct. 9: Download a trace from PhysioBank (see Matlab page), use the FFT to find its power spectrum, and interpret in words how the power spectrum seems to encode things about the original trace.
Friday, Oct.16: Experiment with how the Fourier series behaves at a discontinuity (there are examples among the Matlab codes). Give numerical evidence that it converges in L2, but not uniformly, i.e., it is always far from its pointwise limit somewhere. You should be able to motivate conjectures about the "sup norm" of the difference between the Fourier series and its pointwise limit, and also about the way the L2 norm of that difference approaches zero.
Friday, Oct. 23: Click here for this week's problem.
Friday, Oct. 30: Click here for this week's problem.
Friday, Nov. 6: Click here for this week's problem.
Friday, Nov. 13: Computations with Bessel functions.