
786 SECOND-ORDER ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES
◮ Domain: 0 ≤ x ≤ a, 0 ≤ y ≤ b. Second bound ary value problem.
A rectangle is considered. Boundary conditions are prescribed:
∂
x
w = f
1
(y) at x = 0, ∂
x
w = f
2
(y) at x = a,
∂
y
w = f
3
(x) at y = 0, ∂
y
w = f
4
(x) at y = b.
Solution:
w(x, y)=−
A
0
4a
(x−a)
2
+
B
0
4a
x
2
−
C
0
4b
(x−b)
2
+
D
0
4b
y
2
+K
−b
∞
X
n=1
A
n
λ
n
cosh
nπ
b
(a−x)
cos
nπ
b
y
+b
∞
X
n=1
B
n
λ
n
cosh
nπ
b
x
cos
nπ
b
y
−a
∞
X
n=1
C
n
µ
n
cos
nπ
a
x
cosh
nπ
a
(b−y)
+a
∞
X
n=1
D
n
µ
n
cos
nπ
a
x
cosh
nπ
a
y
,
where K is an arbitrary constant, and the coefficients A
n
, B
n
, C
n
, D
n
, λ
n
, and µ
n
are
expressed as
A
n
=
2
b
Z
b
0
f
1
(ξ) cos
nπξ
b
dξ, B
n
=
2
b
Z
b
0
f
2
(ξ) cos
nπξ
b
dξ,
C
n
=
2
a
Z
a
0
f
3
(ξ) cos
nπξ
a
dξ, D
n
=
2
a
Z
a
0
f
4
(ξ) cos