I'm taking a course in partial differential equations, we have been taught that there are 3 classifications of PDE:
- Parabolic
- Hyperbolic
- Elliptic
Id struggling with understanding why PDEs are separated into these 3 categories. I know that by finding the discriminant of the principal part of the PDE (the coefficients of the highest order terms) you can put an equation in a category, but why does it fall into this category & what does it mean geometrically(if applicable)? and what does it mean when a PDE falls into multiple categories in different parts of the region its defined in?
I would also like to know why, for example, applying a hyperbolic method to a parabolic PDE (even if its the same PDE in a different part of its region) will fail.