7.4 Justifying the series solution
Theorem 7.4.1. Let \(g\) be piecewise continuous on \([0, L]\). Then the series (7.1) converges for \(t > 0\), together with all its term-by-term derivatives, uniformly on \([0, L] \times [t_0, \infty )\) for each \(t_0 > 0\). Its sum \(u\) is \(C^\infty \) for \(t > 0\), satisfies the heat equation and the boundary conditions, \(\int _0^L(u(x, t) - g(x))^2\,dx \to 0\) as \(t \to 0\), and \(u(x, t) \to g(x)\) at every point of continuity where \(g\) is piecewise \(C^1\).
Proof. \(|b_k| \leq \frac 2L\int _0^L|g| =: B\). A term differentiated \(i\) times in \(x\) and \(j\) times in \(t\) is bounded by \(B(k\pi /L)^{i + 2j}\kappa ^je^{-\kappa (k\pi /L)^2t_0}\), and these numbers are summable in \(k\), so every differentiated series converges uniformly by the M-test. Each term solves the equation and vanishes at the ends, hence so does \(u\). By Parseval, \(\int _0^L(u - g)^2 = \frac L2\sum b_k^2(1 - e^{-\kappa \lambda _kt})^2\), which tends to \(0\) by dominated convergence for series, since \(\sum b_k^2 < \infty \). The pointwise statement is Abel’s theorem applied to the convergent series of Theorem 7.2.2, with the decreasing factors \(e^{-\kappa \lambda _kt}\). □
Uniqueness, from the maximum principle of Section 6 or from energy, shows that the series is the solution. For the wave equation the situation is different: the series converges only as well as the series of the data, and to get a classical solution the data must be smooth enough for \(\sum k^2(|a_k| + |\beta _k|/k) < \infty \); otherwise the series defines the weak solution of d’Alembert’s formula.
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