1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

4.5 Potential in a Disk 275

15.Find product solutions of the potential equation in the half-planey>0:

∂^2 u

∂x^2

+∂

(^2) u
∂y^2
= 0 , −∞<x<∞, 0 <y<∞,
u(x, 0 )=f(x), −∞<x<∞.
What boundedness conditions mustu(x,y)satisfy?
16.Convert your solution of Exercise 15 into the following formula (see Ex-
ercise 8 of Section 4.4 and Section 2.11):
u(x,y)=

1

π

∫∞

−∞

f(x′)

y

y^2 +(x−x′)^2 dx

′.

17.Use the formula in Exercise 16 to solve the potential problem in the upper

half-plane, with boundary condition

u(x, 0 )=f(x)=

{ 1 , 0 <x,

0 , x<0.

18.Solve the problem stated in Exercise 15 if the boundary function is

f(x)=

{

1 , |x|<a,

0 , |x|>a.

19.Show thatu(x,y)=xis the solution of the potential equation in a slot

under the boundary conditionsf(x)=x,g 1 (y)=0,g 2 (y)=a. Can this

solution be found by the method of this section?

4.5 Potential in a Disk

If we need to solve the potential equation in a circular diskx^2 +y^2 <c^2 ,itis
naturaltousepolarcoordinatesr,θ, in terms of which the disk is described
by 0<r<c. We found in Section 4.1 that the potential equation in polar
coordinates is
1
r

∂

∂r

(

r

∂v

∂r

)

+

1

r^2

∂^2 v

∂θ^2 =^0.

There are some special features of this coordinate system. First, it is clear that
some coefficients of the Laplacian are negative powers ofr. Thus, we must
enforce a boundedness condition atr=0. Second,θandθ+ 2 πrefer to the
same angle. Therefore, we must require that the functionv(r,θ)be periodic
with period 2πinθ.