Basic PDEs

Get started with the heat equation

Let's see how heat diffuses over time

$\pd{T}{t}=D_T \nabla^2 T$

Explore the wave equation

Play with waves and vibrations

$\pdd{u}{t}=D \nabla^2 u$

Sources and sinks of heat

Diffusion in an inhomogeneous medium

$\pd{T}{t}= \vnabla\cdot(g(x,y)\vnabla T)+f(x,y)$

Inhomogeneous waves

Waves in an inhomogeneous medium

$\pdd{u}{t}= \vnabla\cdot(f(x,y)\vnabla u) $

Schrödinger equation

Quantum mechanics

$i\hbar\pd{\psi}{t}=-\frac{\hbar}{2m}\nabla^2 \psi+V(x,t)\psi$

Bending beams and deforming plates

The plate equation

$\pdd{u}{t}=-D^2 \nabla^4 u-Q$

Convection–diffusion

Movement along streamlines

$\pd{u}{t}=D \nabla^2 u-\v{v}\cdot \vnabla u$