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