Understanding the mathematical nature of PDE
 

Hello everyone,
I am fledgling in the domain. I wanted to learn how to identify the mathematical nature of PDE, whether they are hyperbolic/parabolic/elliptic in space and time. I was taught how to make such categorization for a PDE in 2 independent space variables, but how do we go about it when a time derivative is present? And how does understanding the nature of equation and finding characterisitics further help in our errand of solving the equation numerically? I have read some books about it, but I was not able to understand the physical meaning or visualize how exactly the conclusions were drawn :confused:
Any help would be appreciated.
Regards,

I suggest using classification by means of eigenvalue analysis. This leads to understand the nature of characteristic curves and classify the PDE

Ok, so after calculating the eigenvalues of the the matrix operating on the dependent variable, i am able to identify whether its hyperbolic or elliptic or hybrid etc... but is there a physical explanation of why those equations behave in that preset manner when determining domain of influence or dependence? The books had just mentioned how the domain of influence or dependence changes with the number of characterisitcs emerging from the point, but can we explain it physically without having to go for mathematical proof, because it would be easier to visualize the phenomenon that way.
thanks:)

yes, see the book of Zucrow for examples