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 |
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Last modified: Nov. 25, 1998