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Jack Baskin School of EngineeringUC Santa Cruz

Homework - Winter 2011

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Homework, Winter 2011

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:
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
  • To be handed in: 3 graphs containing the solution of the heat equation in a rectangle of size LxH, with diffusivity k, sides held at T=0, as in the lecture notes. The solutions must be plotted at times t=0, t = tdiff/2 and t = tdiff, where tdiff = (L2 + H2)/k. You may choose any initial condition T(x,y,0) you want, except 0. In your answer, indicate clearly your selected values of k, L, H, and initial condition.
  • Examples of Maple files, to help you with the syntax
  • 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
  • TAKEHOME MIDTERM. To be handed in, in class, on Thursday February 10th
Week 6
  • Write a formal solution to the diffusion equation with arbitrary forcing, say, F(x,t), and no-flux boundary conditions, on the Cartesian interval [0,L] (e.g. the "drunks exiting the pub problem").
  • 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.
Week 7
  • No homework for next week. Please review all of the 2nd order PDE material, and prepare questions if you have any. As a suggestion, you can start looking at more problems for past-year finals. You should be able to do any of the questions on 2nd order PDEs at this point.
Week 8
Week 9