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**,

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,

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

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

Therefore,

Next,

Therefore,

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

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

Therefore,

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

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

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

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

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

So Fourier probably thought: If one writes:

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

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

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