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