188 Chapter 2 The Heat Equation
EXERCISES
1.Find the explicit form forv(x)in terms of the function in Eq. (8) assuming
a. α 1 =β 1 =0, c 1 =c 2 =0;
b. α 1 >0orβ 1 >0, and no coefficient negative.
Why are these two cases separate?
2.Justify each of the conclusions.
3.Derive the general form ofu(x,t)if the boundary conditions are∂u/∂x= 0
at both ends. In this case,λ^2 =0isaneigenvalue.2.10 Semi-Infinite Rod
Up to this point we have seen only problems over finite intervals. Frequently,
however, it is justifiable and useful to assume that an object is infinite in length.
(Sometimes this assumption is used to disguise ignorance of a boundary con-
dition or to suppress the influence of a complicated condition.) Thus, if the rod
we have been studying is very long, we may treat it assemi-infinite—thatis,as
extending from 0 to∞. If properties are uniform and there is no “generation,”
the partial differential equation governing the temperatureu(x,t)remains
∂^2 u
∂x^2=^1
k∂u
∂t, 0 <x, 0 <t.Let us suppose that atx=0 the temperature is held constant, say,u( 0 ,t)= 0
in some temperature scale. In the absence of another boundary, there is no
other boundary condition. However, it is desirable thatu(x,t)remain finite —
less than some fixed bound — asx→∞.
Thus, our mathematical model is
∂^2 u
∂x^2 =
1
k∂u
∂t,^0 <x<∞,^0 <t, (1)
u( 0 ,t)= 0 , 0 <t, (2)
u(x,t) bounded asx→∞, (3)
u(x, 0 )=f(x), 0 <x. (4)The heat equation (1) and the boundary condition (2) are homogeneous.
The boundedness condition (3) is also homogeneous in an important way:
A (finite) sum of bounded functions is bounded. Thus, we can attack Eqs. (1)–
(3) by separation of variables. Assume thatu(x,t)=φ(x)T(t),sothepartial