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Supplementary Problems

Solution: We equate the rate of change of thermal energy in the segment (α . . . β)
with the heat entering the segment through the endpoints.

β
Z
φ cρA dx = k(β, φ(β))A(β)φ (β, t) − k(α, φ(α))A(α)φ (α, t)
x
x
t
α
β
Z
β
φ cρA dx = [kAφ ]
t
x α
α
β β
Z Z
φ cρA dx = (kAφ ) dx
t
x x
α α
β
Z
cρAφ − (kAφ ) dx = 0
t
x x
α
Since the domain is arbitrary, we conclude that
cρAφ = (kAφ ) .
t
x x

9.2 Supplementary Problems

Exercise 183. Solve the initial boundary value problem for a heat equation:
a) u = 25u , u(x, 0) = x(1 − x), u(0, t) = u(l, t) = 0.
t
xx

l
x, 0 ≤ x ≤


b) u = u , u(x, 0) = l 2 , u(0, t) = u(l, t) = 0.
t
xx
l − x, ≤ x ≤ l


2
Exercise 184. Separate variables for the equation tu = u +2u with the bound-
t
xx
ary conditions u(0, t) = u(π, t) = 0. Show that there are an infinite number of
solutions that satisfy the initial condition u(x, 0) = 0. So uniqueness is false for this
equation!
Exercise 185. Consider a metal rod (0 < x < l), insulated along its sides but
not at its ends, which is initially at temperature = 1. Suddenly both ends are
plunged into a bath of temperature = 0. Write the differential equation, boundary
conditions, and initial condition. Write the formula for the temperature u(x, t) at

later times. In this problem, assume the infinite series expansion

4 πx 1 3πx 1 5πx
1 = sin + sin + sin + . . . .
π l 3 l 5 l

Exercise 186. A quantum-mechanical particle on the line with an infinite poten-
´
tial outside the interval (0, l) (”particle in a box”) is given by Schr’odinger’s equation

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