The
Four Main PDEs

Equation/ Property |
Transport Equation |
Laplace/Poisson |
Heat Equation |
Wave Equation |

Physical Interpretation |
transport of mass; flux
proportional to the density u |
diffusion process with no time
dependence (or for large times) |
diffusion process; flux
proportional with the gradient |
wave propagation; linear
elasticity (force proportional to the gradient) |

Regularity | same as initial data |
Analytic solution |
C^{\infty} solutions |
Same as initial data |

Exact/Fundamental Solution |
Exact solution |
Fundamental solution for R^n Green's Functions for some domains |
Fundamental solution for R^n |
D'Alembert's formula (n=1) Fundamental solutions in odd and even dimensions |

Speed of Propagation | finite |
N/A |
infinite |
finite |

Energy |
Conserved in time (for constant coefficients) |
Minimized by the solution (Dirichlet's Principle) | Dissipates in time |
Conserved in time |

Other Properties |
Method of Characteristics Duhamel's formula |
Mean Value Theorem Maximum Principle Harnack's Inequality |
Mean Value Theorem Maximum Principle Backward uniqueness of solutions Duhamel's formula |
Characteristic cone Huygens Principle Duhamel's formula |