§3 Standard Examples

You must know by heart:

• The Laplace equation. Steady heat conduction in a plate:

  

  Also describes ideal flows, unidirectional flows, membranes, electro and magnetostatics, complex functions, ...
  In any number of dimensions: .
  Properties:
  • Smooth solutions.
  • Boundary-value problems.
  • Maximum property.
  • Unlimited region of influence.
  • A simple example of an elliptic equation.
• The heat equation. Unsteady heat conduction in a bar:

  

  Also describes unsteady unidirectional flow, ...
  In any number of dimensions .
  
  • Smooth solutions.
  • Initial/boundary value problems.
  • Maximum property.
  • Unlimited region of influence in space.
  • A simple example of a parabolic equation.
• Vibrations of a string: the wave equation:

  

  Also describes acoustics in a pipe, steady supersonic flow, water waves, optics, ...
  In any number of dimensions .
  
  • Propagating waves.
  • Propagates singularities.
  • Initial/boundary value problems.
  • Energy conservation.
  • Finite propagation speed.
  • A simple example of a hyperbolic equation.

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