1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 5 483

The first three nonzero terms are, fora=b, those with(m,n)=

( 1 , 1 ), ( 1 , 3 )=( 3 , 1 ), ( 3 , 3 ). All terms with an even index are 0.

u(a/ 2 ,a/ 2 ,t)∼=

16 T

π^2

(

e−^2 τ−

2

3 e

− 10 τ+^1

9 e

− 18 τ

)

,

whereτ=ktπ^2 /a^2.

5.u(r)=

(

a^2 −r^2

)

/2andu(r)=

∑∞

1

CnJ 0 (λnr), withCn=^2 a

2

αn^3 J 1 (αn)

.

7.w(x,t)=a 0 +

∑∞

n= 1

ancos(λnx)exp

(

−λ^2 nkt

)

,

v(y,t)=

∑

bmsin(μmy)exp

(

−μ^2 mkt

)

,

whereμm=mπ/b,λn=nπ/a, and initial conditions are

v(y, 0 )= 1 , 0 <y<b; w(x, 0 )=Tx/a, 0 <x<a.

9.J 0 (λr)exp(−λ^2 kt).

11.Bk=bk/k(k+ 1 )fork= 1 , 2 ,...;b 0 must be 0, andB 0 is arbitrary.

13. 
    

((

1 −x^2

)

y′

)′

− m

2

1 −x^2 y+μ

(^2) y=0.
15.u(r,z)=

∑∞

n= 1

ansinhsinh(λ(λnz)

nb)

J 0 (λnr),

whereλn=αn

a

andan=^2 U^0

αnJ 1 (αn)

.

17.u(r,z,t)=sin(μz)J 0 (λr)sin(νct)is a product solution ifμ=mπ/b,λ=
αn/a,andν=

√

μ^2 +λ^2. The frequencies of vibration are thereforeνc

or

c

√(

mπ

b

) 2

+

(

αn

a

) 2

.

19. Each of the two terms satisfies∇^2 φ=−( 5 π^2 )φ.Ony=0andx=1,
    both terms are 0; ony=xthey are obviously equal in value, opposite in
    sign.


20. Each term satisfies∇^2 φ=−( 16 π^2 / 3 )φ.

Ony=0,φ=sin( 2 nπx)−sin( 2 nπx);

ony=

√

3 x,φ=sin( 4 nπx)+ 0 −sin( 2 nπ· 2 x);