So I was studying the numerical analysis of Convection-Diffusion equation. It said that it's a Hyperbolic equation in this form $$\frac{\partial u}{\partial t}+div(u\beta)+\sigma u=0$$ where $$\beta=\mbox{Velocity Field ,} \sigma=\mbox{Absorption coeffecient}$$
My question is how it is Hyperbolic? $B^2-4AC=0$, then how are we saying its Hyperbolic?
Reference: Numerical Solution of PDE using FEM, Claes Johnson, Pg: 168
The classification you're using is only valid for second-order equations and does not apply to first-order equations such as the one you're considering. Take a look at the wikipedia page in the section on hyperbolic systems and conservation laws. Your equation is of this form where the "system" is degenerate in the sense that $s=1$ (in wikipedia's notation), i.e. the equation is scalar. Nevertheless, if you apply the definition used there you will find that your problem is hyperbolic.