1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 3 463

c.w(x,t)=

∑

cnφn(x)e−λ^2 nkT, φn(x)=e−μx/^2 sin(nπx/L),

λ^2 n=

(nπ

L

) 2

+μ

2

4

;

d.λ^2 n=( 7. 30 n^2 + 0. 0133 )× 10 −^4 m−^1.

35. a. ∂u
    ∂t

=D∂

(^2) u
∂x^2
,0<x<L,0<t,
∂u
∂x(^0 ,t)=0, u(L,t)=S^0 ,0<t;
u( 0 ,t)=0, 0 <x<L;
b. u(x,t)=S 0 +

∑∞

n= 1

cncos(λnx)exp

(

−λ^2 nDt

)

,

cn= 4 S 0 (− 1 )n/( 2 n− 1 ).

37.T(y,t)= 300 − 150 y/c+

∑

bnsinλn(y+c)exp(−λ^2 nkt),λn=nπ/ 2 c,

bn=(400 cos(nπ)+ 1000 )/nπ. c. Just before timet=0, the three terms

addto0.Justaftertimet=0, the integrated terms do not change sensi-

bly, but in the first term, neary=c,T(y,t)changes suddenly.

Chapter 3

Section 3.1

1. [u]=L,[c]=L/t.

3.v(x)=(x

(^2) −ax)g
2 c^2.
Section 3.2
3.u(x,t)=

∑∞

n= 1

bnsin

(nπx

a

)

sin

(nπct

a

)

,

bn=

2 a( 1 −cos(nπ))

n^2 π^2 c.

5.u(x,t)=

∑∞

n= 1

ancos

(

nπct

a

)

sin

(

nπx

a

)

,an= 2 U 01 −cosnπ(nπ/^2 ).

7. a. sin

(nπx

a

)

;b.sin

( 2 n− 1

2

πx

a

)

.