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18.3 Dirichlet Problem for a Disk 647

There remains to evaluate the integrals:
1 π 81
2
2
81cos (ξ)sin (ξ)dξ = π,
2π −π 4

π 0 if n = 4

2 2
81cos (ξ)sin (ξ)cos(nξ)dξ =
−81π/8 if n = 4,
−π
and
π

2
2
81cos (ξ)sin (ξ)sin(nξ)dξ = 0.
−π
The solution is
r
1 81π 1 81π 4
U(r,θ) = − cos(4θ)
2π 4 π 8 3
81 1 4
= − r cos(4θ).
8 8
To convert this solution to rectangular coordinates, use the fact that
2
4
cos(4θ) = 8cos (θ) − 8cos (θ) + 1
to obtain
81 1 4 4 4 2 4
U(r,θ) = − (8r cos (θ) − 8r cos (θ) +r )
8 8
81 1 4 4 2 2 2 4
= − (8r cos (θ) − 8r r cos (θ) +r ).
8 8
Then
81 1
2
4
2
2 2
2
2
u(x, y) = − (8x − 8(x + y )x + (x + y ) )
8 8
81 1
2
4
4
2
= − (x + y − 6x y ).
8 8
SECTION 18.3 PROBLEMS
In each of Problems 1 through 8, write the solution of the In each of Problems 9 through 12, solve the problem by
Dirichlet problem for the disk, with the given boundary converting it to polar coordinates.
data.

1. R = 3, f (θ) = 1 9. ∇ u(x, y)= 0for x + y < 16,
2
2
2
2. R = 3, f (θ) = 8cos(4θ) u(x, y)= x for x + y = 16
2
2
2
2
3. R = 2, f (θ) = θ − θ 10. ∇ u(x, y)= 0for x + y < 9,
2
2
2
4. R = 5, f (θ) = θ cos(θ) u(x, y)= x − y for x + y = 9
2
2
5. R = 4, f (θ) = e −θ 11. ∇ u(x, y)= 0for x + y < 4,
2
2
2
2
6. R = 1, f (θ) = sin (θ) u(x, y)= x − y for x + y = 4
2
2
2
2
7. R = 8, f (θ) = 1 − θ 2 12. ∇ u(x, y)= 0for x + y < 25,
2
2
2
8. R = 4, f (θ) = θe 2θ u(x, y)= xy for x + y = 25
2
2
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October 14, 2010 15:27 THM/NEIL Page-647 27410_18_ch18_p641-666