Mathematics 464
Differential Equations II
(aka, Partial Differential Equations)
Spring 2002
Ref. #
1
2
3.
4.
5.
6.
7.
8.
9
10.
11.
12.
13.
14.
15.
Problem
Problem 1.1
Suppose that a metal plate is placed in the first quadrant with corners at (0, 0) , (3, 0) ,
(3, 2) , and (0, 2) . The upper and lower edges of the plate are kept at the constant
temperature of 40 , and the right edge of the plate is kept at a constant temperature of
100 . The left edge of the plate is insulated. The initial temperature distribution in the
plate varies linearly from 10 at the bottom edge of the plate to 60 at the top edge.
Let u ( x, y, t ) denote the temperature at position ( x, y ) at time t .
(a) Give the PDE together with the boundary conditions that u ( x, y, t ) must satisfy.
(b) Give the PDE together with the boundary conditions for the steady state distribution
of temperature.
(c) Describe the symmetry involved in the steady state temperature distribution.
2.5.1 (odds only)
2.5.12
3.2.2
3.3.2
4.3.1 to 4.3.4 all
4.3.9
4.3.10
4.4.1, 4.4.2 and find the Fourier series for these functions. In all three cases, graph the
function f ( x) , along with the partial sums ( n  5, 10, 20 ) of its Fourier series, Fourier
cosine series, and Fourier sine series.
4.4.3 and 4.4.4
Show that total energy is constant for the wave equation with Dirichlet BC’s. Hint:
Follow the procedure done in class and make a clear and convincing argument that
E '(t )  0 .
5.2.4 & 5.2.5
5.3.2 (parts 2 & 3 only) Use Mathematica. Define the important constants like k
(which is 4 in this exercise, but may be different in a different problem), L , etc., and
the initial condition f ( x) . In your submission, turn in plots of the solution for
t  0,1, 2, 4 . Once I have approved your writtin solution, plot the solution for
t  0(0.2)5 , animite the graphs and show them to me, either in the lab just after class,
or come by my office.
5.4.2 (parts 1 & 3 only) Use Mathematica. Define the important constants like c
(which is 4 in this exercise, but may be different in a different problem), L , etc., and
the initial position f ( x) and velocity g ( x ) . In your submission, turn in plots of the
solution for t  0,1, 2 . Once I have approved your writtin solution, plot the solution for
t  0(0.1)2 , animite the graphs and show them to me, either in the lab just after class,
or come by my office.
16.
17.
5.5.3 (part 1 only) Use Mathematica. Define the important constants like c, L, H , etc.,
and the initial condition f ( x, y ) which you can take to be f ( x, y )  xy ( L  x) rather
than the piecewise definition stated in the problem. In your submission, turn in plots
(3D surface plots) of the solution for several values of t . Once I have approved your
writtin solution, plot the solution for more values of t , animite the graphs and show
them to me, either in the lab just after class, or come by my office.
5.6.3 Hint: Use the result for u3 that we derived in class, i.e., don’t rederive it. Use
symmetry to get the solution u1 . Then go through the derivation of the solution for
u2 . Finally use symmetry to get the solution for u4 .
18.
Use Mathematica to solve Laplace’s equation on the rectangle 0, 2  0,3 with
19.
boundary conditions u ( x, 0)  x , u ( x,3)  0 , u  0, y   2 y , u  2, y   0 . Plot the
solution (a 3D plot) on the rectangle.
Prove the properties of Laplace Transforms found at the top of p. 423.
Remember to write Ref. # xx at the top of your sheet. Also, begin each problem on a new sheet
of paper.