An introduction to partial differential equations 527
from which, X = A cos px + B sin px
and Y = Ce py + De − py
or Y = C cosh py + D sinh py
or Y = E sinh p(y + φ)
Hence u(x, y) = X Y
= {A cos px + B sin px}{E sinh p(y + φ)}
or u(x, y)
= {P cos px + Q sin px}{sinh p(y + φ)}
where P = AE and Q = BE.
The first boundary condition is: u(0, y) = 0, hence
0 = P sinh p(y + φ) from which, P = 0.
Hence, u(x, y) = Q sin px sinh p(y + φ).
The second boundary condition is: u(1, y) = 0, hence
0 = Q sin p(1) sinh p(y + φ) from which,
sin p =0, hence, p =nπ for n =1, 2, 3, ...
The third boundary condition is: u(x, 0) = 0, hence,
0 = Q sin px sinh p(φ) from which,
sinh p(φ) = 0 and φ =0.
Hence, u(x, y) = Q sin px sinh py.
Since there are many solutions for integer values of n,
u(x, y) =
∞
n=1
Q n sin px sinh py
=
∞
n=1
Q n sin nπ x sinh nπ y
(a)
The fourth boundary condition is: u(x, 1) = 4 = f (x),
hence, f (x) =
∞
n = 1
Q n sin nπ x sinh nπ(1).
From Fourier series coefficients,
Q n sinh nπ = 2 × the mean value of
f (x) sin nπ x from x = 0 to x = 1
i.e. =
2
1
1
0
4 sin nπ x dx
= 8
−
cos nπ x
nπ
1
0
= −
8
nπ
(cos nπ − cos 0)
=
8
nπ
(1 −cos nπ)
= 0 (for even values of n),
=
16
nπ
(for odd values of n)
Hence, Q n =
16
nπ(sinh nπ)
=
16
nπ
cosech nπ
Hence, from equation (a),
u(x,y) =
∞
n=1
Q n sin nπ x sinh nπ y
=
16
π
∞
n(odd)=1
1
n
(cosech nπ sin nπ x sinhnπy)
Now try the following exercise
Exercise 203 Further problems on the
Laplace equation
1. A rectangular plate is bounded by the
lines x = 0, y = 0, x = 1 and y = 3. Apply the
Laplace equation
∂
2 u
∂x 2 +
∂
2 u
∂ y 2 = 0 to determine
the potential distribution u(x, y) over the plate,
subject to the following boundary conditions:
u =0 when x = 0 0≤ y ≤ 2,
u =0 when x = 1 0≤ y ≤ 2,
u =0 when y = 2 0≤ x ≤ 1,
u =5 when y = 3 0≤ x ≤ 1.
⎡
⎣ u(x, y) =
20
π
∞
n(odd)=1
1
n
cosechnπ sin nπ x sinh nπ(y −2)
⎤
⎦
2. A rectangular plate is bounded by the
lines x = 0, y = 0, x = 3, y = 2. Determine the
potential distribution u(x, y) over the rectangle using the Laplace equation
∂ 2 u
∂x 2 +
∂ 2 u
∂ y 2 = 0, subject to the following
boundary conditions:
u(0, y) = 0
0≤ y ≤ 2,
u(3, y) = 0
0≤ y ≤ 2,
u(x, 2) = 0
0≤ x ≤ 3,
u(x, 0) = x(3 − x) 0 ≤ x ≤ 3.
⎡
⎣ u(x, y) =
216
π 3
∞
n(odd)=1
1
n 3 cosech
2nπ
3
sin
nπ x
3
sinh
nπ
3
(2 − y)
⎤
⎦