Procedure

The steps to take to choose the functions and find the coefficients are:

1.

Make the boundary conditions for the eigenfunctions homogeneous
For heat conduction in a bar, this means that if nonzero end temperatures or heat fluxes through the ends are given, you will need to eliminate these.
Typically, you eliminate nonzero boundary conditions for the eigenfunctions by substracting a function u₀ from u that satisfies these boundary conditions. Since u₀ only needs to satisfy the boundary conditions, not the partial differential equation or the initial conditions, such a function is easy to find.
If the boundary conditions are steady, you can try substracting the steady solution, if it exists. More generally, a low degree polynomial can be tried, say u₀=A+Bx+Cx², where the coefficients are chosen to satisfy the boundary conditions.
Afterwards, carefully identify the partial differential equation and initial conditions satisfied by the new unknown u-u₀. (They are typically different from the ones for u.)

2.

Identify the expansion functions f_n
To do this substitute a single term C_n f_n into the homogeneous partial differential equation. Then take all terms involving f_n and the corresponding independent variable to one side of the equation, and C_n and the other independent variables to the other side. (If that turns out to be impossible, the P.D.E. cannot be solved using separation of variables.)
Now, since the two sides of the equation depends on different coordinates, they must both be equal to some constant. The constant is called the eigenvalue.
Setting the f_n-side equal to the eigenvalue gives an ordinary differential equation you get the eigenfunctions f_n from. In particular, you get the complete set of eigenfunctions f_n by finding all possible solutions to this ordinary differential equation. (If the ordinary differential equation problem for the f_n turns out to be a regular Sturm-Liouville problem of the type described in the next section, the method is guaranteed to work.)
The equation for the C_n is usually safest ignored. They probably taught you in your undergraduate classes to also solve for the C_n, but this only works for homogeneous P.D.E.s. If you insist on solving it instead of what is recommended here, please remember that the eigenfunctions f_n do not have undetermined constants, but the coefficients C_n do. It are the undetermined constants in C_n that allow you to satisfy the initial conditions. They probably did not make this fundamental difference between the functions f_n and the coefficients C_n clear in your undergraduate classes.
There is one case in which you do need to use the equation for the C_n: in problems with more than two independent variables, where you want to expand the C_n themselves in a generalized Fourier series. That would be the case for the pipe wall without axial symmetry. Simply repeat the above separation of variables process for the P.D.E. satisfied by the C_n.

3.

Find the coefficients
Now find the Fourier coefficients C_n (or C_mn for three independent variables) by putting the Fourier series expansion into the P.D.E. and initial conditions.
While doing this, you will also need to expand the initial condition and any inhomogeneous term in the P.D.E. into a Fourier series of the same form. You can find the coefficients of these Fourier series using the orthogonality property described in the next section.
You will find that the P.D.E. produces ordinary differential equations for the individual coefficients. And the integration constants in solving those equations follow from the initial conditions.
(For homogeneous P.D.E.s, the ordinary differential equations will be the same as those found in the previous step.)

Afterwards you can play around with the solution to get other equivalent forms. For example, you can interchange the order of summation and integration (which results from the orthogonality property) to put the result in a Green's function form, etcetera.

11/15/00 0:05:24