Chapter 2 459
Section 2.8
1.x=∑∞
n= 1cnφn,1<x<b;cn= 2 nπ^1 −bcos(nπ)
n^2 φ^2 +ln^2 (b).
3. 1=
∑∞
n= 1cnφn,0<x<a;cn= 2 nπ^1 −ea/ (^2) cos(nπ)
n^2 π^2 +a^2 / 4
.
(Hint: Find the sine series ofex/^2 .)5.bn=∫rlf(x)ψn(x)p(x)dx.- 1 and
√
2cos(nπx),n= 1 , 2 ,....Section 2.9
- a.v(x)=constant; b.v(x)=AI(x)+B.
- If∂u/∂x=0 at both ends, then the steady-state problem is indeterminate.
But Eqs. (1)–(3) are homogeneous, so separation of variables applies di-
rectly. Note thatλ 0 =0andφ 0 =1. The constant term in the series for
u(x,t)is
a 0 =∫r
l∫pr(x)f(x)dx
lp(x)dx.
Section 2.10
- The solution is as in Eq. (9), withB(λ)= 2 T(cos(λa)−cos(λb))/λπ.
3.u(x,t)is given by Eq. (6) withB(λ)=π(α^22 T+^0 λλ (^2) ).
5.u(x,t)=
∫∞
0A(λ)cos(λx)exp(
−λ^2 kt)
dλ;A(λ)=^2 πλT(
sin(λb)−sin(λa))
.
7.u(x,t)=T 0 +∫∞
0B(λ)sin(λx)exp(
−λ^2 kt)
dλ;B(λ)=2
π∫∞
0(
f(x)−T 0)
sin(λx)dx.- a.v(x)=C 0 e−ax;
b.∂w
∂t =D(∂ (^2) w
∂x^2 −a
(^2) w
)
,0<x,0<t,