Classifying a 2nd Order PDE: Understanding the Significance of the Discriminant

[Diophantus]

A quick question:

When classifying a 2nd order PDE as either Hyperbolic, Parabolic or Elliptic we look at whether the discriminant is either positive, zero or negative respectively. Right. What do we do if the discriminant depends on independent variables (or the dependent variable for that matter) such that its sign can vary? Eg D = x. Do we classify it for the different values of x?

Regards.

 

[saltydog]

A quick question:

When classifying a 2nd order PDE as either Hyperbolic, Parabolic or Elliptic we look at whether the discriminant is either positive, zero or negative respectively. Right. What do we do if the discriminant depends on independent variables (or the dependent variable for that matter) such that its sign can vary? Eg D = x. Do we classify it for the different values of x?

Regards.

Yep, yep. Here's a quote:

"If the coefficients A, B, C are functions of x, y, and/or u (dep. variable), the equation may change from one classification to another at various points in the domain".

 

[HallsofIvy]

And, in fact, there are entire books written on "Hyperbolic-Elliptic" equations, "Parabolic-Elliptic" equations, etc.