Introduction

Classification groups P.D.E. with similar properties together.

Example: inviscid fluid flow past a wing cross-section:

Subsonic region M<1: smooth solutions, unlimited region of dependence, iterative solution. These features are indicative of an elliptic PDE.

Supersonic region M>1: singularities, characteristic lines, spatial marching solution. Typical for a hyperbolic PDE.

One set of PDEs that has a unambiguous classification are 2D second order quasilinear equations:

a u_xx + 2 b u_xy + c u_yy + d = 0

where a=a(x,y,u,u_x,u_y), b=b(x,y,u,u_x,u_y), c=c(x,y,u,u_x,u_y), and d=d(x,y,u,u_x,u_y).

The classification for these equations is:

• b² - ac > 0: hyperbolic
• b² - ac = 0: parabolic
• b² - ac < 0: elliptic