The Laplace Par tial Differential Equation 281
Again, the arbitrary constants A(n) and B(n) can be evaluated by taking advantage of the
orthonormality of the y-dependent eigenfunctions, with respect to the weight function
w(y) = 1 over the interval I ={y |0 <y<b}. The statement of orthonormality reads
b
0
Y
n
(y)Y
m
(y)dy = δ(n, m) (5.3)
for n, m = 0, 1, 2, 3,....
The boundary conditions on the problem determine whether the solution is written in terms of
the eigenfunctions with respect to x or y. Remember that the decision comes from our seeking
that particular variable that experiences the homogeneous boundary conditions.
DEMONSTRATION: We seek the steady-state temperature distribution in a thin rectangular
plate over the rectangular domain D ={(x,y) |0 <x<1, 0 <y<1}. The lateral surfaces of
the plate are insulated. The sides y = 0 and y = 1 are insulated. The side x = 1 has a fixed
temperature of zero, and the side x = 0 has a temperature distribution f(y) given as follows.
SOLUTION: The Laplace partial differential equation is
∂
2
∂x
2
u(x, y) +
∂
2
∂y
2
u(x, y) = 0
The boundary conditions are
u(0,y)= f(y) and u(1,y)= 0
and
u
y
(x, 0) = 0 and u
y
(x, 1) = 0
We seek that variable whose boundary conditions are homogeneous. Here, we see that the
homogeneous boundary conditions are with respect to the y variable. Thus, by the method of
separation of variables, we write the two ordinary differential equations as
d
2
dx
2
X(x)−λX(x) = 0
and
d
2
dy
2
Y(y) +λY(y) = 0
Because both ends along the y-axis are insulated, the corresponding homogeneous boundary
conditions on the y equation are type 2 at y = 0 and type 2 at y = 1.
Y
y
(0) = 0 and Y
y
(1) = 0