Transport
The initial profile translates rigidly.
uₜ+cuₓ=0 ⇒ u(x,t)=g(x−ct)
Introduction to Partial Differential Equations
Propagation and diffusion across space and time
PDEs describe fields whose values change across both space and time.
watchone idea
→manipulateone example
→leave withone intuition
The essential idea: transport shifts structure without distortion, while diffusion spreads and smooths it.
One Concept · One Example
Introduction to Partial Differential Equations
Presented by Charlotte Moser · Coming soon
What to notice
- Transport speed and direction are set by c.
- Heat diffusion conserves total mass on an infinite domain.
- The PDE form determines how information propagates.
The mathematical structure
Heat equation
A point impulse becomes a widening Gaussian.
uₜ=νuₓₓ, u=(4πνt)⁻¹ᐟ²e^{−x²/(4νt)}
Contrast propagation with diffusion
Switch between the transport and heat-equation examples from the slides and animate their spatial profiles.
initial profile g(x)translated profile
Transport carries information; diffusion spreads and smooths it.