Problem Set 2 – Due on Feb. 27, 11:59 PM.
Problem 1 – Getting to know Bessel functions better We encountered Bessel’s functions
when solving for the temperature distribution in a half-space (z > 0,⇢ ! 0) due to a
disk of radius a at constant temperature T1. You are asked to answer a few questions here
in order to get to know the Bessel functions better.
The Bessel equation is given by:
x2 d2y
dx2 + x
dy
dx
+ (x2 − ⌫2)y = 0 (1)
The solutions to the above equation for integer values of ⌫ are J⌫(x) and Y⌫(x), Bessel’s
functions of order ⌫ of the first (J) and second (Y ) kinds.
1. Use DLMF (http://dlmf.nist.gov) to determine the asymptotic forms of J⌫(x) and
Y⌫(x) as x # ⌫. You should see that J⌫(x) and Y⌫(x) indeed are counterparts of cos(x)
and sin(x) in cylindrical coordinates.
2. Plot the actual functions and their asymptotic versions. Also plot the absolute value
of the relative error on a log-log scale.
3. Show that Zn(x) and Zm(x) are orthogonal if n 6= m. Here, Z can be either J or Y ,
i.e., show that
Z1
0
xJ⌫(#x)J⌫(μx)dx = 0 if # 6= μ. Read chapter 2 of Hanh and Ozisik
to derive this relation.
Problem 2: Heat is generated uniformly within a 2D rectangular domain of width 2W
and height 2H at a rate of q000 gen W.m−3. Find the temperature distribution for the sides of
the rectangle are held at T0.
Problem 3: A circular region of radius a on the z = 0 plane of half-space occupying
z > 0 is heated by uniformheat flux of magnitude q00 0 W.m−2. If the part ⇢ > a of the surface
z = 0 can be assumed as insulated, find the temperature distribution within the half-plane.
The temperature T ! T0 as
p
⇢2 + z2 # a. Find the average temperature of the circular
disk. Find the e↵ective thermal resistance.
The temperature distribution is still given by T(⇢, z) − T0 =
Z1
0
d⇢e−”zJ0(#⇢)f(#). The
goal is to find the appropriate f(#). To solve this problem, the boundary conditions are:
− k
@T
@z
####
z=0
=
(
q00 0 if ⇢ < a
0 if⇢ > a
(2)
This is infact a slightly easier equation than what we solved in class, where the boundary
condition was on T for ⇢ < a and
@T
@z
####
z=0
for ⇢ > a. To solve this equation, use the identity 