192 Chapter 2 The Heat Equation
∂^2 u
∂x^2=^1
k∂u
∂t, 0 <x, 0 <t,
∂u
∂x( 0 ,t)= 0 , 0 <t,
u(x, 0 )=f(x), 0 <x.5.Determine the solution of Exercise 4 iff(x)is the function given in Exer-
cise 1.
6.Penetration of heat into the earth. Assume that the earth is flat, occupying
the region 0<x(so thatxmeasures distance down from the surface). At
the surface, the temperature fluctuates according to season, time of day, etc.
We cover several cases by taking the boundary condition to beu( 0 ,t)=
sin(ωt), where the frequencyωcanbechosenaccordingtotheperiodof
interest.
a. Show thatu(x,t)=e−pxsin(ωt−px)satisfies the boundary condition
and is a solution of the heat equation ifp=√
ω/ 2 k.
b. Sketchu(x,t)as a function oftforx= 0 ,1, and 2 m, takingω= 2 ×
10 −^7 rad/s (approximately one cycle per year) andk= 0. 5 × 10 −^6 m^2 /s.
c. Withωas in partb, find the depth (as a function ofk)atwhichseasons
are reversed.
7.Consider the problem∂^2 u
∂x^2 =1
k∂u
∂t,^0 <x,^0 <t,
u( 0 ,t)=T 0 , 0 <t,
u(x, 0 )=f(x), 0 <x.Show that, for our method of solution to work, it is necessary to haveT 0 =
limx→∞f(x). Find a formula foru(x,t)if this is the case.
8.If the separation constant in Eq. (5) were positive (say,p^2 ), we would at-
tempt to solveφ′′−p^2 φ=0 subject to the conditions, Eq. (7). Solve the
differential equation, and show that any nonzero, bounded solution is not
0atx=0 and that any solution that is 0 atx=0 is not bounded.
9.R.C. Bales, M.P. Valdez and G.A. Dawson [Gaseous deposition to snow,
2: Physical-chemical model for SO 2 deposition,Journal of Geophysical Re-
search, 92 (1987): 9789–9799] develop a mathematical model for the trans-
port of SO 2 gas into snow by molecular diffusion. The governing partial