Math 111.01/02 (Applied Mathematical Analysis I)

Fall 1998

Plan for Week 14

Second-order partial differential equations are classifed according to combinations of their coefficients as parabolic, hyperbolic, or elliptical. The details of this classification are not important here -- suffice it to say that this is analogous to classifying second-degree curves as parabolas, hyperbolas, or ellipses. The heat equation, ut = a2uxx, is parabolic, and the wave equation, utt = a2uxx, is hyperbolic. To complete our brief "survey" of partial differential equations, this week we take up Laplace's equation, uxx + uyy = 0, which is elliptical. It is natural to think of the independent variables x and y in this case as both representing space dimensions (rather than one space variable and one time variable). For example, Laplace's equation represents the steady state (after time has ceased to be an issue -- i.e., ut = 0) of a diffusion process, such as a two-dimensional heat flow. Laplace's equation also models certain kinds of potential functions (e.g., electric, gravitational), so it is sometimes called the potential equation.

As with our other examples, many functions satisfy Laplace's equation -- interesting problems arise when we specify other conditions. Since time is not an issue, there is no reason to have initial conditions. Instead, we specify boundary conditions on a particular domain in the xy-plane. (This is really only a difference in point of view -- our mixed initial/boundary conditions for the heat and wave equations were specified on the boundary of a particular region in the tx-plane.) In particular, if we specify the value of u at each point of the boundary, we have a problem called the Dirchlet problem. This week we study the Dirichlet problem on a rectangle and on a circle.

Our lab this week builds on the two preceding labs, revisitng the heat equation and incorporating what we have learned about Fourier series as building blocks for constructing solutions.

Here is the syllabus for Week 14:

Week 14 Date Topic Reading Activity
M 11/30 Laplace's equation 10.7a
(to p. 607)
Start Take-Home part
of Final Exam
W 12/2 Dirichlet Problem
for a circle
10.7b
F 12/4 The heat equation 10.5a
(to p. 583)
Lab: The Heat Equation
         
     
                    
         
                    


Notes

  1. Your last homework papers will be turned in on Monday, December 7. Those papers should include solutions to all problems in the assignment below, as well as those in the Week 13 assignment. The assignment dates are start dates.
  2. Some of your exercises call for contour plots. You can get these in Maple by using plot3d. After drawing the surface, click on the plot, go to the Style menu, and select Contour. This will draw level curves on the 3d surface. To see the contour map in a plane, change the view to look straight down the vertical axis.
  3. In general, no homework problem will be given full credit unless you have written an explanation of why you know it is correct.
  4. Submit your Week 14 lab report (the Maple file) via e-mail by the end of the day Wednesday, December 9. This will be the last lab report.
  5. Remember to submit your e-mail journal entry on Friday, December 4. This will be the last required entry.
  6. The take-home part of the Final Exam will be handed out on Monday, November 30. Unlike your other take-homes, this will consist of a single essay question with a maximum page limit. The essay is due at the last class meeting, December 9. It will be graded and returned at the "regular" exam time (Dec. 16) -- at which time you will be 50% finished with your exam.
  7. The rest of the Final Exam will take place on Wednesday, December 16, from 7 to 10 pm. Note that this exam is scheduled in the "math block" time, not by class meeting time. It will not, however, be a "block exam" -- that is, it will be different from the exam taken by other sections.

Assignments


David A. Smith <das@math.duke.edu>

Last modified: Nov. 25, 1998