The Four Main PDEs

Transport Equation

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
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