I'm having problems classifying the inviscid burgers equation into elliptic, parabolic or hyperbolic:
$u_t+uu_x=f(x,t)$
Since the classification for 2nd order partial differential equations of the form:
$a_{11}(x,y)u_{xx}(x,y) + 2a_{12}(x,y)u_{xy}(x,y)+a_{22}(x,y)u_{yy}(x,y) + b_{1}(x,y)u_{x}(x,y) + b_{2}(x,y)u_{y}(x,y) + c(x,y)u(x,y) = d(x,y)$
takes place defining:
$\Delta=a_{12}^2-a_{11}a_{22}$
And studying its sign so:
$\Delta=0 \to$ parabolic equation
$\Delta>0 \to$ hyperbolic equation
$\Delta<0 \to$ elliptic equation
I guessed this equation would be parabolic since there aren't any second order derivatives so:
$a_{11}=a_{12}=a_{22}=0$
But checking in the internet it is said to be a hyperbolic equation, and I don't have a clue why it is that way.
Can someone help me? Thank you very much.
Your equation is of the first order. You cannot use the second order condition to check whether it is hyperbolic/parabolic/elliptic.
You can read about the classification of first order equations/systems in this pdf:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=2ahUKEwjGnO_I4JngAhWTHDQIHWy2DE4QFjAEegQIBRAC&url=http%3A%2F%2Fpeople.3sr-grenoble.fr%2Fusers%2Fbloret%2Fenseee%2Fmaths%2Fenseee-maths-IBVPs-3.pdf&usg=AOvVaw3iWOWyyz7PyfihM3IzGIxH