Math 339 Problems

Assignment 21, due by end of exam period: Heat equation revisited.

Assignment 20, due Dec. 7: Verify two properties of the Green's function of the heat equation.

Assignment 19, due Dec. 4: An easy version of the Wiener-Khinchin theorem.

Assignment 18, due Dec. 2: An improper numerical integral

Assignment 17, due Nov. 30: Jacobi polynomials

Assignment 16, due Nov. 18: Warmup for finding a nice basis for functions on the sphere.

Assignment 15: Read "Gordon/Webb on the Shape of a Drum" from the RESOURCES on the ELLA page. Then follow the recipes of that paper to construct the two regions of their Fig. 15, and in the way that is shown in an example there, verify all the relations to be checked in Fig. 17. Be ready to discuss!

Assignment 14: Due Wednesday, Nov. 11: show that the right hand side of Eq. 16 in the Intro to Sturm-Liouville Theory notes is finite. (That finishes the proof that the Bessel series converges to represent square integrable functions f.)

Assignment 13: Due Wednesday, Nov. 4: Click here for Wednesday's problem

Assignment 12: Due Monday, Nov 2: Click here for Monday's problem

Assignment 11: Due Monday, Oct 26: Click here for Monday's problem

Assignment 10: Due Wednesday, Oct. 21: Click here for Wednesday's problem

Assignment 9: Due Monday, Oct. 19. Click here for Monday's problem.

Assignment 8: Due Wednesday, Oct. 7 Here is the recipe for the CONVOLUTION of two functions f and g: take the Fourier transforms of each of them, multiply corresponding Fourier coefficients to get new Fourier coefficients that are products of the original ones, and take the inverse Fourier transform of the result. It is almost exactly like the cross-correlation of two functions from class this morning (Monday), except that you don't take the complex conjugate . of one of the Fourier coefficients Use this definition to find a formula for the convolution in the sense of the FFT, and identify it as a Riemann sum for a certain integral involving f and g (introduce appropriate factors of N to make this identification).

Assignment 7: Due Monday, Oct 5

Fill in the computation that leads to Eq. 23 (Bessel's inequality) in the ELLA notes on Inner Product.

Assignment 6: Due Wednesday, Sept. 30

In class we solved the wave equation on a string secured at x=0 and x=L, given an initial shape that was a kind of narrow Gaussian pulse at rest. The solution was TWO Gaussian pulses, each of half the height, one moving left and one moving right, that reflected inverted from the ends of the string. Read Chapter 17 and explain how this behavior illustrates the D'Alembert solution to the wave equation. Chapter 17 describes an infinite string, but when we use the FFT we are also, in fact, describing an infinite string, with a periodic wave. [Note: I see Chapter 20 is a good reference for today's material]

Assignment 5: Due Monday, Sept. 28

Read Chapter 4 on the derivation of the heat equation, and do Ch. 4, Prob. 3, on p. 31 -- i.e., derive the heat equation in case the thermal conductivity depends on position. Also read Chapter 16, on the derivation of the wave equation.

Assignment 4: Due Wednesday, Sept. 23, beginning of class

Find the Fourier coefficients with respect to the basis ein&pi x of the function on the interval [-1,1] which is 1 on [-1,0] and -1 on [0,1]. How is this representation related to the term-by-term derivative of the Fourier series representation of the triangle function on [0,1]?

A possible solution to Assignment 4.

Assignment 3: Due Friday, Sept 18, beginning of class

Find the Fourier coefficients (with respect to the basis fn(x)=sin(n&pix)) of the "square wave" function on the interval [0,1] that is 0 in the subinterval [0,1/3), 1 in the subinterval [1/3,2/3], and 0 in the subinterval (2/3,1]. Don't hesitate to try doing it different ways, as a check!

Assignment 2: Due Wednesday, Sept 16, beginning of class

The problem below refers to a heat conducting bar on the interval [0,L].

Assignment 1: Due Monday, Sept 14, beginning of class

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