7 Separation of Variables and Fourier Series

On a bounded interval, the heat and wave equations can be solved by looking for solutions of the special form \(X(x)T(t)\). Each such solution is a single mode, a sine profile decaying or oscillating in time, and the general solution is a superposition of modes. To fit the initial data one must expand an arbitrary function in a series of sines and cosines. Fourier claimed in 1807 that this is always possible; making the claim precise, and finding in what sense the series converges, occupied analysts for a century and created much of modern analysis. This section gives the method and the convergence theorems that justify it.

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