7.1 Separation of variables

Consider the heat equation on a rod with ends held at temperature \(0\): \[ u_t = \kappa u_{xx} \quad (0 < x < L,\ t > 0), \qquad u(0, t) = u(L, t) = 0, \qquad u(x, 0) = g(x) . \] Substituting \(u = X(x)T(t)\) and dividing by \(\kappa XT\) gives \(T'/\kappa T = X''/X\). The left side depends on \(t\) only and the right on \(x\) only, so both equal a constant \(-\lambda \): \[ X'' + \lambda X = 0, \quad X(0) = X(L) = 0; \qquad T' = -\kappa \lambda T . \] The boundary value problem for \(X\) has non-zero solutions only for the eigenvalues \(\lambda _k = (k\pi /L)^2\), \(k = 1, 2, \ldots \), with eigenfunctions \(X_k = \sin (k\pi x/L)\); this is the simplest Sturm–Liouville problem, whose general theory is part of the theory of ordinary differential equations. Then \(T_k = e^{-\kappa \lambda _kt}\), and superposing, \begin {equation} \label {eq:heatseries} u(x, t) = \sum _{k = 1}^\infty b_ke^{-\kappa (k\pi /L)^2t}\sin \frac {k\pi x}{L} , \end {equation} with \[ b_k = \frac 2L\int _0^Lg(x)\sin \frac {k\pi x}{L}\,dx , \] the formula for \(b_k\) coming from setting \(t = 0\) and using the orthogonality \(\int _0^L\sin \frac {j\pi x}{L}\sin \frac {k\pi x}{L}\,dx = \frac L2\delta _{jk}\).

For the vibrating string \(u_{tt} = c^2u_{xx}\) with fixed ends, the same separation gives \(T'' = -c^2\lambda _kT\), so each mode oscillates with angular frequency \(\omega _k = k\pi c/L\): \[ u(x, t) = \sum _{k = 1}^\infty \Bigl (a_k\cos \frac {k\pi ct}{L} + \frac {L\beta _k}{k\pi c}\sin \frac {k\pi ct}{L}\Bigr )\sin \frac {k\pi x}{L} , \] with \(a_k\) and \(\beta _k\) the sine coefficients of \(u(x, 0)\) and \(u_t(x, 0)\). The frequencies \(kc/2L\) are the harmonics of the string; that they are integer multiples of the fundamental is why a string sounds a definite pitch. The two solutions differ in one telling way: in the heat series the \(k\)th term is damped by \(e^{-\kappa k^2\pi ^2t/L^2}\), so high modes die almost at once; in the wave series nothing is damped, and the smoothness of \(u\) is exactly that of the data.

With insulated ends, \(u_x = 0\), the eigenfunctions are \(\cos (k\pi x/L)\), \(k = 0, 1, \ldots \), and the solution is a cosine series whose \(k = 0\) term, the mean of \(g\), is the limit as \(t \to \infty \). With periodic conditions (a ring) both sines and cosines appear, and the data must be expanded in a full Fourier series.

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