Test 2 Review — Math 401-501/502 (Spring 2013)

General Information

Calculators: You may use scientific calculators to do numerical calculations — logs, exponentials, and so on. You may not use any calculator that has the capability of doing algebra or calculus, or of storing course material.

Other devices: You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Formulas: You will be given a table of Fourier transforms (Table A.2, plus a couple more), and a few integrals.

Coverage: The test will be over material covered from after Test 1 on through Thursday, 4/18/13. This includes chapter 2, sections 5.1, 5.2, 5.3 (Laplace equation in a circular disk, p. 105), 5.5 (Vibrating circular membrane, p. 117), 8.1.

Topics Covered

• Fourier series
  1. Be able to find Fourier series, Fourier sine series, and Fourier sine series for various functions. Be able to use symmetry to calculate Fourier coefficients.
  2. Fourier's convergence theorem. Be able to define piecewise continuous and piecewise smooth functions. Know how to extend functions — 2L periodic extensions; 2L even periodic extensions; 2L odd periodic extensions. Be able to graph the functions the series converge to, and know the three types of convergence that we discussed in class (notes, 3/28) and for homework (see assignments 6 and 7).
     1. Pointwise convergence — jumps; Gibbs' phenomenon.
     2. Uniform — only corners, no jumps; visual.
     3. Mean convergence — energy of residual goes to 0; only need finite energy to begin with.
  3. Parseval's identity — assignments 6 and 7.

• Separation of variables
  1. Be able to separate variables to solve the heat equation in a bar of finite length, the wave equation for a finite string, and the potential equation in a disk. See assignment 8.
  2. Be able to solve eigenvalues problems associated with separation of variables. Be able to obtain the Rayleigh quotient for these eigenvalue problems. Here is an example of how to do that. Consider this eigenvalue problem: $\ X''+ \lambda X = 0$, $X'(0)=0$, $X(1)=0$. Do the following steps.
     1. Multiply by the equation by $X$:   $XX''+ \lambda X^2=0$.
     2. Integrate: $\int_0^1 XX''dx + \lambda \int_0^1 X^2dx =0$
     3. Integrate the first integral by parts and use the boundary conditions: \[ \int_0^1 XX''dx = XX'\bigg|_0^1 - \int_0^1 X'^2 dx =\underbrace{X(1)}_{0} X'(1) - X(0)\underbrace{X'(0)}_{0} - \int_0^1 X'^2 dx = - \int_0^1 X'^2 dx \]
     4. Substitute this for the integral on the right in step (b) and solve for $\lambda$. The result is the Rayleigh quotient for the problem, \[ \lambda = \frac{\int_0^1 X'^2 dx}{\int_0^1 X^2 dx}. \]
  3. Be able top obtain the normal modes, in terms of Bessel functions, for the vibrating drumhead. Follow my notes from 4/16/13 and 4/18/13. Be able to derive the Rayleigh quotient in the this case. I will write up notes recapping how to do this. (In the text, section 5.5, p. 117, is good, but it covers more material than I want to.)

• Fourier transforms
  1. Know how to find the Fourier transform or inverse Fourier transform of simple functions. These will be similar to the ones in assignment 9.
  2. Be able to use Fourier transform techniques to solve problems similar to those for a heat equation in an infinite bar and an infinite vibrating string. Again, see assignment 9. Be able to derive the heat kernel. Know what the error function is and be able to use it to solve heat flow problems, given the solution in terms of the heat kernel.

Updated: April 17, 2013 (fjn)