208 Chapter 2 The Heat Equation
u( 0 ,t)=T 0 ,0<t,
u(x, 0 )={ 0 , 0 <x<a,
T 0 , a<x.13.∂^2 u
∂x^2 =1
k∂u
∂t,0<x<∞,0<t,
∂u
∂x(^0 ,t)=0,^0 <t,
u(x, 0 )={
T 0 , 0 <x<a,
0 , a<x.- ∂
(^2) u
∂x^2 =
1
k∂u
∂t, −∞<x<∞,0<t,
u(x, 0 )=exp(
−α|x|)
, −∞<x<∞.15.∂^2 u
∂x^2 =1
k∂u
∂t, −∞<x<∞,0<t,u(x, 0 )={ 0 , −∞<x<0,
T 0 , 0 <x<a,
0 , a<x<∞.16.∂^2 u
∂x^2 =1
k∂u
∂t,0<x<a,0<t,
∂u
∂x(^0 ,t)=0, u(a,t)=T^0 ,0<t,
u(x, 0 )=T 0 +S(a−x),0<x<a.
17.Give a physical interpretation for this problem and thus explain why
u(x,t)should increase steadily astincreases. (Assume thatSis a pos-
itive constant.)
∂^2 u
∂x^2=^1
k∂u
∂t, 0 <x<a, 0 <t,
∂u
∂x( 0 ,t)= 0 , ∂u
∂x(a,t)=S, 0 <t,
u(x, 0 )= 0 , 0 <x<a.18.Show thatv(x,t)=(S/ 2 a)(x^2 + 2 kt)satisfies the heat equation and the
boundary conditions of the problem in Exercise 17. Also findw(x,t),
defined byu(x,t)=v(x,t)+w(x,t).
19.Show that the four functionsu 0 = 1 , u 1 =x, u 2 =x^2 + 2 kt, u 3 =x^3 + 6 kxt