288 Chapter 4 The Potential Equation
18.For the potential problem on an annular ring
1
r∂
∂r(
r∂(^2) u
∂r
)
+r^12 ∂(^2) u
∂θ^2 =^0 , a<r<b,
show that product solutions have the form
A 0 +B 0 ln(r), (C 0 +D 0 θ)ln(r),
or
rn
(
Ancos(nθ)+Bnsin(nθ))
+r−n(
Cncos(nθ)+Dnsin(nθ))
.
19.Solve the potential problem on the annular ring as stated in Exercise 18
with boundary conditions
u(a,θ)= 1 , u(b,θ)= 0.20.Find product solutions of the potential equation on a sector of a disk
with zero boundary conditions on the straight edges.
∇^2 u= 0 , 0 ≤r<c, 0 <θ <α,
u(r, 0 )= 0 , u(r,α)= 0.21.Solve the potential problem in a slit disk:
∇^2 u= 0 , 0 ≤r<c, 0 <θ< 2 π,
u(r, 0 )= 0 , u(r, 2 π)= 0 ,
u(r,θ)=f(θ ), 0 <θ< 2 π.22.Show that the functionu(x,y)=sin(πx/a)sinh(πy/a)satisfies the po-
tential problem
∇^2 u=^0 ,^0 <x<a,^0 <y,
u( 0 ,y)= 0 , u(a,y)= 0 , 0 <y,
u(x, 0 )= 0 , 0 <x<a.
This solution is eliminated if it is also required thatu(x,y)be bounded
asy→∞.
23.Solve the potential equation in the rectangle 0<x<a,0<y<b, with
the boundary conditionsu( 0 ,y)= 0 , ∂u
∂x(a,y)= 0 , 0 <y<b,
u(x, 0 )= 0 , u(x,b)=x, 0 <x<a.