Epilogue: Classification

Abstract

In Section 2.1 we indicated that there is a way of grouping second order PDE’sinto classes,such that the solutions of PDE’s in the same class have similar qualitative properties. We shall discuss this classification now and relate it to our physical classification into equations which describe diffusion, wave propagation, and equilibrium situations. We let

$$Lu = au_{xx} + 2bu_{xy} + cu_{yy} + du_x + eu_y + fu$$

((11.1))

be the general second-order operator in two variables with constant coefficients. For convenience in this discussion, we shall assume that a > 0. The characteristic polynomial associated with L is

$$q\left( {x,y} \right) = ax^2 + 2bxy + cy^2 + dx + ey + f.$$

((11.2))