Classification of PDEs

Differential Equations · Axiom Academy

One number — the discriminant B^2-4AC — sorts every second-order linear PDE into elliptic, parabolic, or hyperbolic, and each label predicts the physics. 1. The General Form and the Discriminant A second-order linear PDE in two variables x,y has the general form below, where A , B , C are coefficients (possibly functions of x,y ) and the lower-order terms never affect the classification — only the second-derivative coefficients do. General second-order linear PDE The classification discriminant 2. Elliptic PDEs: Steady-State Equilibrium When , the PDE is elliptic . The defining example is Laplace's equation, which governs steady-state temperature, electrostatic potential, and membrane deflection once everything has settled down and stopped changing in time. When , the PDE is parabolic . The defining example is the heat equation, which governs how a temperature (or concentration, or price) profile evolves forward in one distinguished direction — time. Writing it as A=1 (the u_ xx coefficient), B=0 , C=0 (no u_ yy term), — parabolic. 4. Hyperbolic PDEs: Wave Propagation When , the PDE is hyperbolic . The defining example is the wave equation, which governs disturbances that travel at a finite, fixed speed — sound, light, and vibration all obey this pattern. Here (writing u_ tt -c^2u_ xx =0 in the A u_ xx +Bu_ xy +Cu_ yy form with y=t ), so — hyperbolic.

This is the written version of the interactive lesson above. See the full Differential Equations course.