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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.

TobyStack
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    The preface to Partial Differential Equations by Evans states, "I also find it unsatisfactory to 'classify' partial differential equations: this is possible in two variables, but creates the false impression that there is some kind of general and useful classification scheme available in general." – littleO Nov 17 '17 at 11:06

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