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18.3 Dirichlet Problem for a Disk 645

6. Apply separation of variables to the problem 8. Solve for the steady-state temperature distribution in
2
∇ u = 0for0 < x < a,0 < y < b, a homogeneous, thin, flat plate covering the rect-
angle 0 ≤ x ≤ a,0 ≤ y ≤ b if the temperature
∂u on the vertical and lower sides are kept at zero
u(x,0) = (x,b) = 0for 0 ≤ x ≤ a,
∂y and the temperature along the top side is f (x) =
2
x(x − a) .
u(0, y) = 0,u(a, y) = g(y) for 0 ≤ y ≤ b.
7. Use separation of variables to solve 9. Solve for the steady-state temperature distribution in
a thin, flat plate covering the rectangle 0 ≤ x ≤
2
∇ u = 0for0 < x < a,0 < y < b,
4, 0 ≤ y ≤ 1 if the temperature on the horizontal
u(x,0) = 0,u(x,b) = f (x) for 0 ≤ x ≤ a, sides is zero while the temperature on the left side
is f (y) = sin(πy) and on the right side, g(y) =
∂u
u(0, y) = (a, y) = 0for 0 ≤ y ≤ b. y(1 − y).
∂x
18.3 Dirichlet Problem for a Disk

We will solve the Dirichlet problem for a disk of radius R centered at the origin. The boundary
of this disk is the circle x + y = R . Using polar coordinates, the problem for u(r,θ) is
2
2
2
2
2
∂ u 1 ∂u 1 ∂ u
2
∇ u = + + = 0for0 ≤r < R,−π ≤ θ ≤ π
∂r 2 r ∂r r ∂θ 2
2
and
u(R,θ) = f (θ) for − π ≤ θ ≤ π.
n
n
It is easy to check that the functions 1,r cos(nθ), and r sin(nθ) are harmonic on the entire
plane. Thinking ahead to the possibility of a Fourier series to satisfy the boundary condition,
attempt a solution in a series of these functions:
∞
1 n n
u(r,θ) = a 0 + (a n r cos(nθ) + b n r sin(nθ)).
2
n=1
This would require that
∞
1
n
n
u(R,θ) = f (θ) = a 0 + (a n R cos(nθ) + b n R sin(nθ)).
2
n=1
n
This is a Fourier expansion of f (θ) on [−π,π] if we choose the entire coefficients, a n R and
n
b n R to be the Fourier coefficients of f on [−π,π]. This means that
1 π
a 0 = f (ξ)dξ
π −π
and, for n = 1,2,···,
1 π 1 π
a n = f (ξ)cos(nξ)dξ and b n = f (ξ)sin(nξ)dξ.
R n −π R n −π
Another form of this solution is
1 π
u(r,θ) = f (ξ)dξ
π −π
1 r π π
∞ n
+ f (ξ)cos(nξ)dξ cos(nθ) + f (ξ)sin(nξ)dξ sin(nθ) .
π R −π −π
n=1
(18.1)

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October 14, 2010 15:27 THM/NEIL Page-645 27410_18_ch18_p641-666