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u = g(x)

u = j(y) u = k(y)
x

u + u = h(x)
y

Figure 10.5: Inhomogeneous boundary mixed conditions in the rectangle

u = g(x)

u = 0 u = 0
x

u + u = 0
y

Figure 10.6: Homogeneous boundary mixed conditions in the rectangle

0
So 0 = Y (0)+Y (0) = Bβ n +A. Without losing any information we may pick B = −1,
so that A = β n , Then
Y (Y ) = β n cosh β n y − sinh β n y
Therefore, the sum

∞
X
u(x, y) = A n sin β n x(β n cosh β n y − sinh β n Y )
n=0
is a harmonic function in D that satisfies all three homogeneous BCs. The remaining
BC is u(x, b) = g(x). It requires that

∞
X
g(X) = A n (β n cosh β n b − sinh β n b) · sinh β n x
n=0

for 0 < x < a. This is simply a Fourier series in the eigenfunctions sin β n x. The
coefficients are given by the formula

Z a
2
A n = (β n cosh β n b − sinh β n b) −1 g(x) sin β n xdx.
a 0
Example 10.2 The same method works for a three-dimensional box {0 < x < a, 0 <
y < b, 0 < z < c} with boundary conditions on the six sides. Take Dirichlet conditions

103