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Supplementary Problems

5.2 Supplementary Problems

Exercise 104. By trial and error, find a solution of the diffusion equation u = u xx
t
2
with the initial condition u(x, 0) = x .

Exercise 105.
(a) Show that the temperature of a metal rod, insulated at the end x = 0, satisfies

the boundary condition ∂u/∂x = 0. (Use Fourier’s law.)
(b) Do the same for the diffusion of gas along a tube that is closed off at the end
x = 0. (Use Fick’s law.)
(c) Show that the three-dimensional version of (a) (insulated solid) or (b) (imper-

meable container) leads to the boundary condition ∂u/∂n = 0.

Exercise 106. A homogeneous body occupying the solid region D is completely
insulated. Its initial temperature is f(x). Find the steady-state temperature that it

reaches after a long time. (Hint: No heat is gained or lost.)

Exercise 107. Suppose that some particles which are suspended in a liquid

medium would be pulled down at the constant velocity V > 0 by gravity in the
absence of diffusion. Assume homogeneity in the horizontal directions x and y.
Let the z axis point upwards. Taking account of the diffusion, find the boundary

condition on an impermeable plane z = a.
Solution: ku = V u on z = a.
z

Exercise 108. Two homogeneous rods have the same cross section, specific heat
c, and density ρ but different heat conductivities κ and κ and lengths L and L .
1
1
2
2
Let k = κ /cρ be their diffusion constants. They are welded together so that the
j
j
temperature u and the heat flux κu at the weld are continuous. The left-hand rod
x
has its left end maintained at temperature zero. The right-hand rod has its right
end maintained at temperature T degrees.
(a) Find the equilibrium temperature distribution in the composite rod.
(b) Sketch it as a function of x in case κ = 2, κ = 1, L = 3, L = 2, and
2
1
2
1
T = 10. (This exercise requires a lot of elementary algebra, but it’s worth it.)
2
Exercise 109. Consider the problem d u + u = 0, u(0) = 0 and u(L) = 0,
dx 2
consisting of an ODE and a pair of boundary conditions. Clearly, the function
u(x) ≡ 0 is a solution. Is this solution unique, or not? Does the answer depend on
L?

0
0
Exercise 110. Consider the problem u”(x) + u (x) = f(x), u (0) = u(0) =
1 [u (l) + u(l)], with f(x) a given function. (a) Is the solution unique? Explain.
0
2
(b) Does a solution necessarily exist, or is there a condition that f(x) must satisfy
for existence? Explain.

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