2.11 Infinite Rod 193
differential equation is∂C
∂t=D
(∂ (^2) C
∂x^2
−a^2 C
)
,
whereCis the concentration of SO 2 as a function ofx(depth into the snow)
and time andDis a diffusion constant. The term containingCappears be-
cause the SO 2 takes part in a chemical reaction with water in the snow,
forming sulphuric acid, H 2 SO 4 .Thecoefficienta^2 depends onpH, temper-
ature, and other circumstances; we treat it as a constant. The problem is to
be solved for a wide range of values for the parameters.
If the snow is deep, the authors believe that it is reasonable to use a semi-
infinite interval forxand to add the conditionC(x,t)→0asx→∞.In
addition, a natural boundary condition at the snow surface is that con-
centration in the snow match that in the air:C( 0 ,t)=C 0 .Furthermore,
if the snow is fresh, we can assume that the concentration throughout is
initially 0,C(x, 0 )= 0 , 0 <x.a.Find a steady-state solutionv(x)that satisfies the partial differential
equation and the boundary conditions.
b.State the problem (partial differential equation, boundary condition at
x=0, condition asx→∞, and initial condition) to be satisfied by the
transientw(x,t)=C(x,t)−v(x).
c. Solve the problem for the transient. Note that the condition asx→∞
must be relaxed to:w(x,t)bounded asx→∞.Individualproductso-
lutions do not approach 0 asxincreases.2.11 Infinite Rod
If we wish to study heat conduction in the center of a very long rod, we may as-
sume that it extends from−∞to∞. Then there are no boundary conditions,
and the problem to be solved is
∂^2 u
∂x^2 =1
k∂u
∂t, −∞<x<∞,^0 <t, (1)
u(x, 0 )=f(x), −∞<x<∞, (2)
∣∣
u(x,t)∣∣
bounded asx→±∞. (3)