Just do the assigned problems.
"Review Problems"
are suggested problems for the TA to do in office hours.
| Week | Problem set | Problems (Unless noted, all problems are from Haberman 3rd Ed.) |
|---|---|---|
| 1 | 1 | 1.4.1d,e, 1.4.2, 1.4.7b,1.2.4 (Review 1.3.2, 1.4.1g,1.4.7a) Solutions (pdf file) (courtesy Y Fang) |
| 2 | 2 | 2.2.3, 2.3.1a,c, 2.3.2a,c,e,f (Review 2.3.2e,g) Solutions (pdf file) |
| 3 | 3 | 2.3.6, 2.3.7, 2.3.11 also show the orthogonality of sin(n*Pi*x/L)cos(m*Pi*x/L) over -L<=x<=L for all n,m Note: (5 Feb)! Do as much of 2.3.11 as you can we haven't exactly covered all of this yet. Solutions (pdf file) |
| 4 | 4 | 2.5.1a, 2.5.6b, 2.5.10 (Review 1.5.3, 2.5.15c ) Solutions (pdf file) |
| 5 | 5 | 3.3.1c, 3.3.2a, 3.3.18, 3.4.4 (Review 3.4.6, 3.4.12) Solutions (pdf file) |
| 6 | 6 | 4.2.1 (note: the last term in Eq. 4.2.7 should be Q(x,t)\rho_0(x)), 4.4.3, 4.4.7 (review 4.4.1) Solutions (pdf file) |
| 7 | 7 | 5.3.3,, 5.3.9 (a new twist: hint the substitution x=exp(y) will be handy), 5.5.1e, and for the regular Sturm-Louiville Boundary conditions given in Eq. 5.3.3 (page 156). (review problems...3.4.11, 5.4.3, 5.4.4 (if you can't do the final integrals, just set the problem up) Solutions (pdf file) |
| 9 | 8 | 7.3.4a, 7.3.6, 7.4.1 (Review 7.3.1d,7.5.7) Revised 20 March 2001 at 6:24pm, sorry for the late swap Solutions (pdf file) |
| 10 | 9 | 7.8.1 (Lot's-o-hints: a,c,e think Rayleigh quotient d frankly I don't know where he get's his answer from. Instead, use matlab (or any other program of your choice) to make a plot of the function in part (b) and estimate the zeros. f now use a root finder (like fzero in matlab) to find the first eigenvalue. Here's an example matlab script for plotting and finding the zeros of J0 that might be useful; 7.9.4b (Hint: you might want to know that d/dz J_0(z)=-J_1(z)), 7.9.5 (Hint: see section 7.10) (Review 7.9.1a,b ) Solutions (pdf file) |
| 11 | 10 | 8.2.1a,d,e, 8.2.2a,b and either 8.2.4 or 8.2.5 ( Review...do the other one) Solutions (pdf file) |
| 12 | 11 | 14 Apr: this is now correct Solve 8.3.7 two ways. First reduce the problem to one with homogeneous boundary conditions (the hint in the back gives an interesting choice of auxiliary function but it's not the only one...if you use it though, show how you find it). Second solve the problem directly using Green's Formula. Also solve 8.6.8: what are the appropriate 3-D eigenfunctions and eigenvalues? Solutions (pdf file) |
| 13 | 12 | Just a few Green's function problems 9.3.6a,b, 9.3.11 and derive the 2-D infinite-space Green's function for Poisson's equation Solutions (pdf file) |
| 14 | 13 | THE END Just a few wave propagation problems for you. 1) Use Fourier transforms to derive the general solution to the 1-D 2nd order wave equation (u_tt=c^2u_xx) on the infinite line with initial conditions u(x,0)=f(x), u_t(x,0)=g(x). (Hint 1: the answer is given on page 536, hint 2: define h(x)=\integral from 0 to x g(y) dy)) 2) some characteristic problems: 12.2.2, (extra credit, do 12.2.4), 12.2.5d, 12.6.6a (note a typo in this problem, u(rho) should be c(rho)). 12.6.19a (Extra credit 12.6.7a which is the same problem but with different initial conditions). hint: for the last 3 problems it is useful to sketch both the characteristics and rho vs x to visualize what is happening. (Review: 12.2.4, 12.3.6, 12.6.9a, 12.6.19d) Solutions (pdf file) |