Mathematical and Computational Methods for Engineers
E155B, Spring 2002
Problem Set #2
(Fourier Transforms, PDEs – Diffusion and Laplace Equations)
Date: 4/10/2002
Due: 4/17/2002
Reading: Kreyszig 10.5, 10.10, 11.1, 11.5
Problem 1 p. 549 Exercise 2
Problem 2 a) Determine the Fourier transform fˆ ( k ) of the following function:
f ( x)  e
a x
defined for all x. Assume a > 0.
b) In the limit as a  0 the original function f (x) approaches unity. Use your result in
part a) to evaluate the Fourier transform of f ( x )  1 . [Hint: you will have to find the area
under fˆ ( k ) in the limit when a  0 ].
Problem 3 p. 608 Exercise 5
Problem 4 a) A rod of length L has its ends A and B kept at 0oC and 100oC respectively
until steady state conditions are reached. At t=0 the temperature of B is suddenly reduced
to 0oC and kept so while the temperature at A is maintained at 0oC. Find the temperature
of the rod T(x,t) at any time t. Express your answer in terms of 2 = K/c.
b) Use MATLAB to plot your solution for L=1 m and =1 m2/sec. Choose four snapshots
in time to adequately represent the temperature transient and plot them on the same set of
axis.
Problem 5 In lecture we have derived a one-dimensional time-dependent heat
conduction equation for a laterally insulated rod. In certain situations, however, as in the
case of a high-power cable on board of a spacecraft, radiative heat transfer and internal
heat generation need to be included as well. Assume the cable to have a shape of a
cylindrical rod of length L and radius r as shown below. Modify our derivation of the
heat equation to include the following:
a) power generation due to Ohmic heating, i.e. dQ = I2dR, where I is the current and
dR is the resistance of an infinitesimal element of length dx
b) radiation from the lateral surface of the cable, assuming dQ = T4dS, where dS is
the lateral surface area of the element. Ignore radiation from the two ends.
L
2r
Express your new PDE for T(x,t) in terms of density , thermal conductivity K, heat
capacity c, electrical conductivity , current I, and radius r.
Problem 6 It was shown in class that the separation constant for a homogeneous timedependent diffusion equation:
 2T T
2 2 
x
t
written in a separated form with T ( x, t )  F ( x) H (t ) as follows

F  1 H 
 2
F
 H
is best expressed as =-k2. Show that the separation constant  cannot possibly be
positive. Multiply both sides of F   F by F and integrate from x=0 to x=l to show that:
l
 F  dx
2
   0l
F
2
dx
0
Explain, based on this result, why  cannot be positive.
[Hint: use integration by parts and assume homogeneous boundary conditions:
F(0)=F(l)=0]
Problem 7 A transmission line 1,000 km long is initially under steady-state conditions,
with potential 1,200 Volts at the source at x=0 and 1,100 Volts at the load (x=1000). The
terminal end of the line is suddenly grounded due to a short, reducing its potential to zero
while the potential at the source is kept at 1,200 Volts. Find the potential along the
transmission line at any time t. Assume negligible inductance.
[Hint: Recall that the potential along a transmission line satisfies the following PDE:
 2V
V
 rC
2
x
t
where r and C are resistance and capacitance per unit length respectively ]
Problem 8 p. 609 Exercise 18 a). Solve the problem analytically first, then use
MATLAB to make a color plot of your solution in two dimensions and a map of lines of
constant temperature (isotherms). [Hint: you may find the following commands: pcolor,
shading, contour together with MATLAB’s Help useful]