2.11 Infinite Rod 197
in the integrand of Eq. (4) will be nearly zero, except for smallλ.Thus,Eq.(4)
is approximately
u(x,t)∼=∫
0(
A(λ)cos(λx)+B(λ)sin(λx))
exp(
−λ^2 kt)
dλfornot large, and the right-hand side may be found to a high degree of
accuracy with little effort.
On the other hand, ifktis small, the exponential in the integrand of Eq. (7)
will be nearly zero, except forx′nearx. The approximation
u(x,t)∼=1
√
4 kπt∫x+hx−hf(x′)exp[−(x′−x) 2
4 kt]
dx′is satisfactory forhnot large, and again numerical techniques are easily applied
to the right-hand side.
The expression in Eq. (7) also has a number of other advantages. It requires
no intermediate integrations (compare Eq. (5)). It shows directly the influence
of initial conditions on the solution. Moreover, the functionf(x)need not
satisfy the restriction
∫∞
−∞∣∣
f(x)∣∣
dx<∞in order for Eq. (7) to satisfy the original problem.
EXERCISES
See the exercise Common Singular Eigenvalue Problems on the CD.
- Find the solution of Eqs. (1)–(3) using the form given in Eq. (7) if the initial
temperature distribution is
f(x)={
T 0 , x<0,
T 1 , 0 <x.- Find the solution of Eqs. (1)–(3) using the form given in Eq. (4) if
f(x)={
T 0
(
a−|x|)
, −a<x<a,
0 , otherwise.- Same task as in Exercise 2, withf(x)=T 0 e−|x/a|for allx.
- Show that the solution of the problem studied in Section 10,
∂^2 u
∂x^2 =1
k∂u
∂t,^0 <x,^0 <t,