Derivation of Fourier's Solution
Fourier began by demonstrating that a stationary solution to the heat
flow problem satisfies the differential equation now known as Laplace's
equation:
(1.1.1.1) 
Pierre
Simon Laplace (1749–1827) and others had come across this
equation in various contexts. In modern terminology, it is simply the
observation that when the flow of heat ( )
has reached a state of equilibrium, it is incompressible ( ).
To solve his partial differential equation (1.1.1.1), Fourier introduced
a technique that is standard today. He searched for special solutions of
the form
When z is of this form, equation (1.1.1.1) reduces to
or, assuming the second derivatives are not zero,
(1.1.1.2) 
The left side of equation (1.1.1.2) is independent of w
while the right side is independent of x.
This implies that both sides are independent of both x
and w,
and so each of these ratios is constant,
Since g(x)
= C g''(x), the sign of g(x)
is either always the same as the sign of g''(x),
or it is always the opposite. If we want z(x,w)
to be continuous, then we need to have
g(–1) = g(1)
= 0, and so g(x)/g''(x)
must be negative:
for some positive constant A. Fourier set  and
solved for  and The coefficient of sin tx
must be zero because g is an even function of x. He then argued
that  must be zero because the temperature will approach
0 as we move away from the source of heat at w =
0. He had found
a solution:
where a and t > 0 are unknown constants. If we want
this solution to be zero at x= ± 1, then t must be an
odd multiple
of  .
The general solution is a sum of such functions:
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