11.07.2016
Dear Math 488 Student:Our class schedule specifies that you should complete Lectures 30 and 31 and the associated homework problems by Sunday, November 13. These are the last lectures and homework that will be covered by Exam 2, which is to be given during the week of November 14 - 18. Peg Pisel will be in touch with you and your exam proctor soon about scheduling Exam 2 and other details.
WEEK 12 - Content Summary for Lectures 30 and 31
Lecture 30 begins with an important solution method for the solution of boundary value problems associated with partial differential equations called the Superposition Principle. The first part of Lecture 30 is devoted to a description of the meaning of superposition in the context of boundary value problems for second order partial differential equations. More specifically: We only consider problems for which the underlying differential equation is the wave equation or the heat equation or Laplace’s equation in two variables.
That may seem too restrictive to allow for a wide variety of interesting and useful applications, but is not, because: Each of these three types of equations but it has a different form for each of the standard coordinate systems in the plane or 3-dimensional space. It turns out that many important applications of boundary value problems in applied mathematics and engineering can be solved by using one of these three types of partial differential equations in one of these coordinate systems.
The Superposition Principle is actually a special property of the set of solutions of linear systems subject to both homogeneous and non-homogeneous boundary conditions. It implies that: If we are given a boundary value problem, call it Problem 1, with only one non-homogeneous boundary condition BC1, and if there is a second boundary value problem, call it Problem 2, with a second non-homogeneous condition BC2, then the boundary value problem subject to both non-homogeneous conditions BC1 and BC2 can be obtained by simply adding together the solutions of Problem 1 and Problem 2! It’s logical and it turns out to be useful in the solution of certain boundary value problems.
We illustrate the Superposition Principle at the beginning of Lecture 31 by going back to the steady-state temperature distribution problem solved in Lecture 29 for a thin metal plate in the xy-plane on the region R = { (x, y) : 0 < x < 1, 0 < y < Pi} subject to the boundary conditions:
u(x, 0) = 0 for 0 < x < 1, u(x, Pi) = 100x(1 - x); u(0, y) = u(1, y) = 0 for 0 < y < Pi.
We also use the Superposition Principle to find the solution of the problem with a second non-homogeneous condition: u)1, y) = 10 Sin[ y].
Next, we model the vibration u(r, t) of a circular drumhead on a region P = {(r, Theta): 0 < r < R, t > 0) that has been struck at the center at t = 0 to centrally symmetric position u(r, 0) = .01+.01 r and initial velocity u sub(t) (r, 0) = 0. When we apply the Method of Separation of Variables, it turns out that equation in r is Bessel's equation subject to the boundary conditions: u(R, 0) = 0 and u(r, t) is a bounded for t > 0.
Unlike the vibrating string analysis, for which the eigenvalues corresponded to a fundamental tone and regular overtones of the string, we find the eigenvalues of the drumhead are irregularly spaced and produce "noise" but not harmonic sound. Part of Lecture 31 works out the mathematical details of this vibration. The rest of Lecture 31 sets up a mathematical model to compute the steady-state temperature distribution on the interior of a solid sphere of radius 1 using Laplace's Equation in spherical coordinates in which the upper surface of the sphere is maintained at 100 degrees while the surface of the lower hemisphere is maintained at 0 degrees.
If you have any questions about this week's work, please contact me.
Important Reminder: Please don’t forget to e-mail me the Progress report requested at beginning of this message!
Have a great week!
Tony Peressini
Cell PH: 217 840 2871
anthonyperessini@gmail.com