10.31.2016
Dear Math 488 Student:Our class schedule specifies that you should complete Lectures 28 and 29 and the associated homework problems by Sunday, November 6. These two lectures begin our discussion of boundary value problems for partial differential equations.
In this course, we do not present a full discussion of the basic solution methods for partial differential equations. That would require a full semester of work even if we limited our discussion to first and second order partial differential equations, which are the most important types from the point of view of applications to science and engineering. Instead, we will focus on three specific types of second-order partial differential equations in two variables, the Wave Equation, the Heat (or Diffusion) Equation, and Laplace's (or the Potential) Equation.
WEEK 11 - Content Summary for Lectures 28 and 29
Lecture 28 begins our discussion of the Wave Equation, which is basic to the solution of problems involving mechanical vibrations. The basic physical model is a string tightly stretched between two fixed posts, such as a piano or guitar string. We show that if the string has length L and if the two ends of the string are along the x-axis at x = 0 and x = L, when the string is plucked at time t = 0, then the displacement u(x,t) of the string from its rest position satisfies the following partial differential equation and boundary conditions:
u sub[tt](x,t) = c^2 u sub[xx] (x, t) for 0 < x < L and t > 0
subject to the boundary conditions u(0, 0) = 0 and u(L, 0) = 0. (*)
We then solve this boundary problem by the Method of Separation of Variables; that is, we look for all solutions of the "product form" u(x, t) = F(x) G(t).
By using the the Method of Separation of Variables we can show that the boundary-value problem (*) reduces to two ordinary differential equations in x and t:
F(x) G''(t) = c^2 F''(t) G(t) for 0 < x < L and t > 0
subject to the conditions: F(0) = 0; F(L) = 0, and G'(0) = 0.(**).
Then we observe that this separated equation in x is a Sturm-Liouville system.
We then proceed to solve these two ordinary differential equations and boundary conditions for different initial displacements u(x, 0) = h(x) and different initial velocities u'(x, 0) = 0.
The corresponding eigenfunction expansions for these systems lead to the Fourier Cosine Series, the Fourier Sine Series and the Full Fourier Series depending on the choice of h(x).
Lecture 28 discusses applications of these eigenfunction expansions to mechanical vibrations and music.
The end of Lecture 28 and all of Lecture 29 carry out a similar analysis of the Heat Equation.
The basic physical model is a metal rod of length L that extends from x = 0 to x = L on the x-axis and that is insulated along its length so that heat flows only in the x-direction. The ends of the rod at x = 0 and x = L are maintained at fixed temperatures.
The steady-state temperature u(x, t) of the rod at time t > 0 is modeled by the heat equation:
u sub{t] (x,y) = c^2 u sub[xx] (x, y) for 0 < x < L and 0 < y < M
subject to boundary conditions on the rectangular region 0 < x < L and 0 < y < M such as one or more of the following:
u(0, y) = A or u(L, y) = B for 0 < x < L or
u(x, 0) = C and u(x, M) = D for 0 < y < M
As in the analysis of the vibrating string problem, we can separate the variables x and y and use the conditions on the boundary of the rectangle:
0 < x < L, 0 < y < M
to obtain (S - L) systems and use the corresponding eigenfunction expansions to solve a variety of steady-state heat flow problems.
As in the case mechanical vibrations, you will see that Fourier series can be used to solve a variety of steady-state heat flow problems. Many examples of such problems are included in the examples in Lecture 29 and in the corresponding homework problems.
I hope that you find these comments helpful as you study these two lectures.
Let me know if you have any questions about these lectures or the associated homework
Have a good week!
Tony Peressini
Cell PH: 217 840 2871
anthonyperessini@gmail.com