Homework, Winter 2012
This homework will help you solidify the bases that you have learnt during the lectures, and test whether you understood them or not. The quiz of the week will be based on one of the problems set.
Answers will be posted regularly. Homework answers provided usually study the problem in much more depth than required, to give you as much material as possible to look at.
Week 1:
- Questions 2.3.1-7
- Scanned questions (just for this once)
- Problems on Fourier Series from HRB: for example: 12.14, 12.16
Week 2
- Questions: 2.4.1a (+ evaluate the coefficients), 2.4.1b (be careful of the n=0 solution), 2.5.1a (hint: be careful what variable has homogeneous BCs), 2.5.15a (read the corresponding part of the book), 3.2.2, 4.4.3, 4.4.7.
Week 3
- Make sure you know the proofs for (a)-(d) theorems for SL theory. (i.e. up to and included "simple eigenvalues")
- Problems 5.3.3, 5.3.9, 5.4.1, 5.4.3 (you don't have to find the eigenfunctions), 5.6.1c
Week 4
- Problems: 5.7.1, 5.8.7
- From the Final of 2010 , Problem 4
- From the Final of 2009 , Problem 2
- From the Final of 2006 , Problem 2
- From the Final of 2005 , Problem 6
Week 5
- Hand in a computer-generated movie of the oscillating drum, with "interesting" initial conditions (not axisymmetric, not a single eigenmode). Email me the movie (any format ok) and the source code.
- Finish any problems you were not able to do last week
- Problem 1 from Final of 2009 (see above)
- Last year's Take-Home midterm. Note: this is not this year's midterm!
Week 6
- No homework this week (Take-home midterm instead)
Week 7
- Write a formal solution to the Poisson equation in a spherical cavity (i.e. the region between two concentric spheres of radius, say, a and b), with arbitraty right-hand-side f(r,theta) and homogeneous Dirichlet conditions.
- Solve the equation (1/r2) d( r2 du/dr) = f(r) with Dirichlet boundary conditions at r=a and r=b using the delta function method. Compare your answer to that of the previous problem when the forcing is only a function of r.
- Problem 1 of Final of 2006
- Problem 4 of Final of 2007
- Problem 9.2.3
- Redo the calculation of the Greens' function we did in class, this time using Laplace transforms. Note: using Laplace transforms on bounded intervals requires a bit of thought... can you see how to do it?
Week 8
- For next week: This homework . Ignore the "textbook" questions.
Week 9
- To be handed in Thursday:
- (a) Problem 2.10 of handout
- (b) Consider a traffic flow problem (see lecture notes) with initial condition u(x,0) =
umax/4 for x<-1/2, u(x,0) = umax/2 for -1/2 < x < 1/2, and u(x,0) =
3umax/4 for x>1/2. This problem has 2 shocks.
- At which points (x,t) do the shocks start?
- Discuss why the two shocks are likely to merge.
- Consider a time t prior to the merger. What are the equations of the shock curves? Solve them to find the positions of the shocks prior to the merger.
- When and where do these two shocks merge?
- What is the evolution of the shock curve after the merger? (What equation does it satisfy, what is the position of the shock, etc..)
- Deduce the complete solution of the problem for all times t and all positions x. Draw the full set of characteristics as well as the shock curves on the same diagram. Draw the function u(x,t) for t=0, for t prior to the shock merger, and for t after the shock merger.