Form of the solution

Before starting the process, you should have some idea of the form of the solution you are looking for. Some experience helps here.

For example, for unsteady heat conduction in a bar of length ,the temperature u would be written

where the f_n are chosen eigenfunctions. The separation of variables procedure allows you to choose these eigenfunctions cleverly.

For a uniform bar, you will find sines and/or cosines for the functions f_n. In that case the above expansion for u is called a Fourier series. In general it is called a generalized Fourier series.

After the functions f_n have been found, the ``Fourier coefficients'' C_n can simply be found from substituting the expression for u in the given P.D.E. and initial conditions. (The boundary conditions are satisfied when you choose the eigenfunctions f_n.)

If the problem was axially symmetric heat conduction through the wall of a pipe, the temperature would still be written

but the expansion functions would now be found to be Bessel functions, not sines or cosines.

For heat conduction through a pipe wall without axial symmetry, the temperature would be written

where the f_n would turn out to be sines and cosines and the g_mn Bessel functions. Note that in the first sum, the temperature is written as a simple Fourier series in ,with coefficients C_n that of course depend on r and t. Then in the second sum, these coefficients themselves are written as a (generalized) Fourier series in r with coefficients that depend on t.

(For steady heat conduction, the coordinate ``t'' might actually be a second spatial coordinate. For convenience, we will refer to conditions at given values of t as ``initial conditions'', even though they might physically really be boundary conditions.)

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