1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 2 461

u(x,t)=T 0 +

∑∞

1

bnsin(λnx)e−λ^2 nkt,

bn=^2 a

∫a

0

(T 1 −T 0 )sin

(

nπx

a

)

dx.

3. SS:v(x)=T 0 +

r

2 x(x−a),0<x<a.

EVP:φ′′+λ^2 φ=0,φ( 0 )=0,φ(a)=0,λn=nπ/a,φn=sin(λnx),

n= 1 , 2 ,....

u(x,t)=T 0 −r 2 x(x−a)+

∑∞

1

bnsin(λnx)exp

(

−λ^2 nkt

)

,

bn=

2

a

∫a

0

[

T 1 −T 0 +

r

2 x(x−a)

]

sin

(nπx

a

)

dx.

5. SS: not needed.
   (Hint: Put−γ^2 uon the other side of the equation. Separation of vari-
   ables givesφ′′/φ=γ^2 +T′/kT=−λ^2 .)
   EVP:φ′′+λ^2 φ=0,φ′( 0 )=0,φ′(a)=0,λ 0 =0,φ 0 =1;λn=nπ/a,
   φn=cos(λnx),n= 1 , 2 ,....
   u(x,t)=e−γ^2 kt

(

a 0 +

∑

ancos(λnx)exp

(

−λ^2 nkt

))

.

a 0 =T 1 /2,an=− 2 T 1

(

1 −cos(nπ)

)

/n^2 π^2.

7.u(x,t)=T 0.

9.u(x,t)=T 0 +

∑∞

n= 1

cnsin(λnx)exp

(

−λ^2 nkt

)

,

λn=(^2 n−^1 )π

2 a

,cn=(T^1 −T^0 )·^4

( 2 n− 1 )π

.

11.u(x,t)=T 0 +

∫∞

0

B(λ)sin(λx)exp

(

−λ^2 kt

)

dλ,B(λ)= −^2 λT^0

π(α^2 +λ^2 )

.

13.u(x,t)=

∫∞

0

A(λ)cos(λx)exp

(

−λ^2 kt

)

dλ,A(λ)=

2 T 0 sin(λa)

πλ.

15.u(x,t)=

∫∞

0

(

A(λ)cos(λx)+B(λ)sin(λx)

)

exp

(

−λ^2 kt

)

dλ,

A(λ)=T^0 sinπλ(λa),B(λ)=T^0 (^1 −πλcos(λa))

or