Computer Algebra Recipes

208 CHAPTER 5. LINEAR PDE MODELS. PART 1
considers a semi-in¯nite solid medium whose bounding planar surface is periodicallyheatedbythesun.
Thedirectiontransversetothesurfaceistakento
be the x-direction, with the surface located at x =0,andpositivextaken to
be inside the surface. In order to animate the temperature pro¯le inside the
surface, Russell loads the plots library package.
> restart: with(plots):
From undergraduate thermodynamics, he knows that the time-dependent temperature
distribution T (x; t) obeys the one-dimensional heat di®usion equation
@T
@t = d @2T ; (5.1)
@x2 with the heat di®usion coe±cient d = K=(½C), where K is the thermal conductivity,
½ the density, and C the speci¯c heat. Russell enters the heat equation.
> heateq:=diff(T(x,t),t)-d*diff(T(x,t),x,x)=0;
³ ´ μ
heateq := @@tT (x; t) ¡ d @ 2
2 T (x; t) =0
@x
To account for the periodic heating of the planar surface due to the sun, he
takes the temperature variation at x =0tobeT (0;t)=T0 cos(!t). Rather
than derive the temperature variation T (x; t) forx > 0, Russell decides to
make an intelligent guess as to its mathematical form and check it by substituting
the form back into the di®usion equation. Since he is looking for a
steady-state response and the temperature should decrease as x increases inside
the surface, Russell conjectures that the solution should be of the structure
T (x; t) =T0 e ¡®x cos(!t¡ ¯x), with the parameters ® and ¯ as yet undetermined.
At x = 0, the boundary condition is satis¯ed, and an exponentially
decaying cosine solution seems reasonable inside the surface. Both ® and ¯
must be positive for a waveform propagating in the positive x-direction.
What forms should ® and ¯ have? From the structure of the heat di®usion
equation, the units of d are m 2 /s, so 1= p d has units of s 1=2 ¢m ¡1 . Since the
argument in the exponential function must be dimensionless, ® must have the
dimensions of m ¡1 . The only other parameter in the problem involving a time
unit is the frequency ! with units s ¡1 . So, noting that the combination p !=d
has the units m ¡1 ,Russelltakes® = a p !=d, wherea is a numerical factor, yet
to be determined. By similar reasoning, the constant ¯ is set equal to b p !=d,
with b another numerical factor. Both a and b must be positive.
To check the postulated solution and determine a and b, Russell enters
T (x; t), which will be automatically substituted into the heat equation.
> T(x,t):=T[0]*exp(-a*sqrt(omega/d)*x)
*cos(omega*t-b*sqrt(omega/d)*x);
T (x; t) :=T0 e (¡a p !
d x) cos
Russell then simpli¯es the heat equation,
> eq:=simplify(heateq);
μ
¡!t+ b
r !
d x