1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Miscellaneous Exercises 357

15.Solve the following potential problem in a cylinder:

1

r

∂

∂r

(

r

∂u

∂r

)

+

∂^2 u

∂z^2 =^0 ,^0 <r<a,^0 <z<b,

u(a,z)= 0 , 0 <z<b,

u(r, 0 )= 0 , u(r,b)=U 0 , 0 <r<a.

16.Find the solution of the heat conduction problem

1

r

∂

∂r

(

r∂u

∂r

)

=^1

k

∂u

∂t

, 0 <r<a, 0 <t,

u(a,z)=T 0 , 0 <t,

u(r, 0 )=T 1 , 0 <r<a.

17.Find some frequencies of vibration of a cylinder by finding product so-
lutions of the problem
1
r

∂

∂r

(

r∂∂ur

)

+∂

(^2) u
∂z^2 =

1

c^2

∂^2 u

∂t^2 ,^0 <r<a,^0 <z<b,^0 <t,

u(r, 0 ,t)= 0 , u(r,b,t)= 0 , 0 <r<a,α<t,

u(a,z,t)= 0 , 0 <z<b, 0 <t.

18.Derive the given formula for the solution of the following potential equa-
tion in a spherical shell:
∇^2 u= 0 , a<ρ<b, 0 <φ<π,
u(a,φ)=f

(

cos(φ)

)

, u(b,φ)= 0 , 0 <φ<π,

u(ρ, φ)=

∑∞

n= 0

An

b^2 n+^1 −ρ^2 n+^1

b^2 n+^1 −a^2 n+^1

(a

ρ

)n+ 1

Pn

(

cos(φ)

)

,

An=^2 n 2 +^1

∫ 1

− 1

f(x)Pn(x)dx.

19.Show that the functionφ(x,y)=sin(πx)sin( 2 πy)−sin( 2 πx)sin(πy)
is an eigenfunction for the triangleTbounded by the linesy=0,y=x,
x=1. That is,
∇^2 φ=−λ^2 φ inT,
φ= 0 on the boundary ofT.
What is the eigenvalueλ^2 associated withφ?