Postgraduate Science

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Partial Differential Equations

Heat, wave, and Laplace equations — the three classical types of linear PDE.

Second-order linear PDEs in 2D classify (via the discriminant $b^2 - ac$) into:

  • Elliptic (Laplace $\Delta u = 0$): steady states; smooth solutions; maximum principle.
  • Parabolic (heat $\partial_t u = \Delta u$): irreversible diffusion; smoothing in time.
  • Hyperbolic (wave $\partial_t^2 u = c^2 \Delta u$): finite propagation speed; sharp wavefronts.

Method of separation of variables: try $u(x,t) = X(x) T(t)$, get ODEs for $X$ and $T$. For the heat equation on $[0, L]$ with Dirichlet BCs the eigenmodes are $\sin(n\pi x/L)$; expanding initial data gives

$$u(x,t) = \sum_n a_n \sin\!\Big(\frac{n\pi x}{L}\Big)\,e^{-n^2\pi^2 t/L^2}.$$

High-frequency modes decay first — the heat equation smooths data instantly. The wave equation preserves amplitude but transports it (d'Alembert's $f(x-ct) + g(x+ct)$).

Interactive: heat diffusion on a 1D rod

Quiz

1. The heat equation $\partial_t u = \alpha \Delta u$ is:
2. The wave equation $\partial_t^2 u = c^2 \Delta u$ propagates information at speed:
3. Solutions to Laplace's equation $\Delta u = 0$ on a domain are:
4. Separation of variables for $u_t = \alpha u_{xx}$ on $[0,L]$ with Dirichlet BCs gives modes:
5. d'Alembert's formula gives wave-equation solutions as:
6. Maximum principle for the heat equation: the maximum of $u$ on $\bar Q$ is attained on: