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18.5 Dirichlet Problem for Unbounded Regions 653

EXAMPLE 18.5
We have a formula for the solution of
2
∇ u(x, y) = 0for x > 0, y > 0,
u(0, y) = 0for y > 0,
and

u(x,0) = 1for x > 0.
With f (x) = 1, we can write the solution
y ∞ 1 y ∞ 1
u(x, y) = dξ − dξ.
2
π 0 y + (ξ − x) 2 π 0 y + (ξ + x) 2
2
A routine integration yields
x
y ∞ 1 1 1
dξ = + arctan
2
π 0 y + (ξ − x) 2 2 π y
and
y ∞ 1 1 1
x
dξ = − arctan .
π 0 y + (ξ + x) 2 2 π y
2
The solution is
2
x
u(x, y) = arctan .
π y
This function is harmonic on the right quarter-plane and u(0, y) = 0for y > 0. Furthermore, if
x > 0,
x
2 2 π
lim arctan = = 1,
y→0+ π y π 2
as required.

SECTION 18.5 PROBLEMS

1. Write an integral solution for the Dirichlet problem for and then derive a solution using an appropriate Fourier
the upper half-plane if transform.
5. Find a general formula for the solution of the Dirichlet
⎧
⎪−1 for −4 ≤ x < 0 problem for the right quarter-plane if u(x,0) = f (x)
⎨
u(x,0) = 1 for 0 ≤ x < 4 and u(0, y) = g(y).
⎪ 6. Write an integral solution for the Dirichlet problem for
0 for |x| > 4.
⎩
the lower half-plane y < 0if u(x,0) = f (x).
2. Write an integral solution for the Dirichlet problem for 7. Find the steady-state temperature distribution in a
the upper half-plane if u(x,0) = e −|x| . thin, homogeneous flat plate extending over the right
3. Write an integral solution for the Dirichlet problem quarter plane if the temperature on the vertical side
for the right quarter-plane if u(x,0) = e −x cos(x) for is e −y and the temperature on the horizontal side is
x > 0and u(0, y) = 0for y > 0. maintained at zero.
4. Write an integral solution for the Dirichlet problem 8. Solve the Dirichlet problem for the strip −∞ < x <
for the right quarter-plane if u(x,0) = 0for x > 0and ∞,0< y <1if u(x,0)=0for x <0and u(x,0)=e −αx
u(0, y) = g(y) for y > 0. Use separation of variables, for x > 0 with α a positive number.

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October 14, 2010 15:27 THM/NEIL Page-653 27410_18_ch18_p641-666