Laplace, heat and wave equations: separation of variables

Wave equation $u_tt = c^{2}u_{xx}$

d'Alembert on $\mathbb{R}: u(x,t) =$ ½$[\varphi(x +$ ct$) + \varphi(x -$ ct$)] + (1/2c)\int_{x-ct}^{x+ct}$ $\psi(s)$ ds.

• Finite propagation speed c; the domain of dependence is [x − ct, x + ct].
• No smoothing: singularities in the data travel forever.
• Energy is conserved; the equation is reversible in time.

On [0, L] with Dirichlet data: $u = \sum (a_{n} \cos(n\pi$ct$/L) + b_{n} \sin(n\pi$ct$/L)) \sin(n\pi{}x/L)$.

Heat equation $u_t = k u_{xx}$

$u = \sum c_{n}$ $e^{-k n^{2}\pi^{2}t/L^{2}}$ $\sin(n\pi{}x/L)$ on [0, L]; on $\mathbb{R}$, convolution with the Gaussian kernel.

• Infinite propagation speed, instant smoothing to $C^\infty$ for t > 0.
• Irreversible: the backward problem is ill-posed.
• Maximum principle: max over the parabolic boundary; no interior max.
• Solutions decay to the steady state as $t \to \infty$.

Laplace $\Delta{}u = 0$

• **Mean value property$**: u(x_{0})$ is the average over any ball/sphere centred at $x_{0}$.
• Maximum principle: no interior max or min on a domain unless constant.
• Harmonic $\Rightarrow C^\infty ($indeed real-analytic).
• Liouville: a bounded harmonic function on $\mathbb{R}^{n}$ is constant.
• Uniqueness: the Dirichlet problem on a bounded domain has at most one solution; the Neumann problem is unique up to an additive constant and needs $\oint \partial{}u/\partial{}n = 0$ for solvability.

Contrast table

	Wave	Heat	Laplace
Speed	finite	infinite	—
Smoothing	none	instant	—
Time reversible	✓	✗	—
Maximum principle	✗	✓	✓

Key takeaways

• d'Alembert for the wave equation on the line; separation of variables on an interval.
• Heat smooths and forgets; the wave keeps everything and travels at speed c.
• Maximum principle holds for heat and Laplace, never for the wave.