Thread: [SOLVED] Partial Differential Equations: How to tell if a PDE is Hyperbolic, Elliptic

1. [SOLVED] Partial Differential Equations: How to tell if a PDE is Hyperbolic, Elliptic

Hey guys,

I am a bit confused as to how to tell if an ordinary differential eqn. is hyperbolic, parabolic, when given a random pde. For example, I was looking at a book and it gave this example to determine the type:

Uxx - 3Uxy = 0

The book said to use det A = (a11*a22) - (a12)^2, where 11 = xx, 12 = xy, 22 = yy. I tried working it out using this method and I got 0-9 = -9, but the book gets -9/4??

I know there is also another method, such as B^2 - 4AC =0, >0, <0. Then thing is, I don't have any clue as to how to extract the B, A, C from the above equation. How can I tell which is A, B, C??? Thanks for any and all help.

2. Originally Posted by spearfish
Hey guys,

I am a bit confused as to how to tell if an ordinary differential eqn. is hyperbolic, parabolic, when given a random pde. For example, I was looking at a book and it gave this example to determine the type:

Uxx - 3Uxy = 0

The book said to use det A = (a11*a22) - (a12)^2, where 11 = xx, 12 = xy, 22 = yy. I tried working it out using this method and I got 0-9 = -9, but the book gets -9/4??

I know there is also another method, such as B^2 - 4AC =0, >0, <0. Then thing is, I don't have any clue as to how to extract the B, A, C from the above equation. How can I tell which is A, B, C??? Thanks for any and all help.
Go back to your notes or text book and check what matrix A is in terms of the coefficients of the PDE (its not what you think).

CB

3. Originally Posted by spearfish
Hey guys,

I am a bit confused as to how to tell if an ordinary differential eqn. is hyperbolic, parabolic, when given a random pde. For example, I was looking at a book and it gave this example to determine the type:

Uxx - 3Uxy = 0

The book said to use det A = (a11*a22) - (a12)^2, where 11 = xx, 12 = xy, 22 = yy. I tried working it out using this method and I got 0-9 = -9, but the book gets -9/4??

I know there is also another method, such as B^2 - 4AC =0, >0, <0. Then thing is, I don't have any clue as to how to extract the B, A, C from the above equation. How can I tell which is A, B, C??? Thanks for any and all help.
In general, a linear second order PDE is of the form

$A u_{xx} + B u_{xy} + C u_{yy} + \,\text{lower order terms}\; = 0$

If $B^2 - 4AC > 0$, then it's hyperbolic,
If $B^2 - 4AC = 0$, then it's parabolic,
If $B^2 - 4AC < 0$, then it's elliptlic.

4. Thanks Danny and CaptainBlack! This helps guide me in the right direction. I appreciate your time.