Fourier Series

The motivation behind this topic is as follows, Joseph-Louis Fourier, (1768-1830), a French engineer (and mathematician) discussed heat flow through a bar which gives rise to the so-called Heat Diffusion Problem, tex2html_wrap_inline111

eqnarray8

where u(0,t)=0=u(L,t), and u(x,0)=f(x), f given. Think of f as being the initial state of the bar at time t=0, and u(x,t) as being the temperature distribution along the bar at the point x in time t. The boundary conditions or conditions at the end-points are given in such a way that the bar's ``ends" are kept at a fixed temperature, say 0 degrees (whatever).

We apply the method of Separation of Variables first, - Fourier assumed that the solution he was looking for had the form,

eqnarray15

where we need to find these two functions f, g. Then,

eqnarray17

Arguing on physical grounds, the bar should reach a steady state as tex2html_wrap_inline135 which means that tex2html_wrap_inline145 is positive, or else g(t) is exponentially large.

Therefore,

eqnarray32

Next,

eqnarray35

Therefore,

eqnarray40

But we still don't know tex2html_wrap_inline145 or the c's! So, Fourier figures the solution looks like,

eqnarray45

Now to use the boundary conditions, ``b.c.", u(0,t) = 0 = u(L,t), we note that this means,

eqnarray50

Therefore,

eqnarray53

since g(t) is not identically equal to zero. But these two conditions on f now determine tex2html_wrap_inline145 , but the tex2html_wrap_inline145 is not unique. The point is that the only solutions of

eqnarray55

which satisfy f(0)=f(L)=0, are those for which

eqnarray58

So there are infinitely many possibilities for tex2html_wrap_inline145 , Each one of these tex2html_wrap_inline173 generates a solution , where

eqnarray67

and these solutions all satisfy u(0,t) = u(L,t) = 0.

But do these solutions satisfy u(x,0)=f(x)?

tex2html_wrap203

So Fourier probably thought: If one writes:

eqnarray73

then u(x,t) also satisfies the conditions u(0,t) = u(L,t)=0, (these are all linearly independent, too).

But, once again, if f(x) is basically arbitrary it is not necessarily true that

eqnarray85

So, the insight was, ``What if f can be represented as an infinite series?'', that is,

eqnarray92

in the sense of convergence of the series on the right to f(x) for each x in [0,L]? It turned out that this could be done! This led to the creation of Fourier Series.

Tue Sep 15 18:38:20 EDT 1998