What are the Different Types of Poisson's Equation?

[Unskilled]

1. Of what type is Poisson’s equation uxx + uyy = f(x,y) ?

I used that if you have auxx+buxy+cuyy+dux+euy +fu+g=0

where a, b, c, d, e, f, g is constants,

and if b^2-4ac<0 then you get an elliptic type because
b=0, a=1, c=1 gives

0^2-4*1*1=-4<0 => elliptic

Is this right? And why do i have to do like this? i don't understand the meaning of type.

 

[HallsofIvy]

Yes, Poisson's equation includes Laplace's equation (with f(x,y)= 0) which is an elliptic equation.

Actually the basic concept of "type" of a partial differential equation is based on the simplified form that you started with. Any equation of the form
Au_xx+ Bu_yy= something is "elliptic", A and B are numbers of the same sign. Any equation of the form Au_xx- Bu_yy= something is "hyperbolic". Any equation of the form Au_xx- Bu_y= something is "parabolic". Having other derivatives just makes things more complicated- but you can always make changes of independent variables to get rid of mixed derivatives and first derivatives. The formula you cite is based on those changes.

The names are, of course, based on the analogy with conic sections.

The distinctions are important because hyperbolic equations always have two "characteristic curves", parabolic equations one, and elliptic equations none.

For example the hyperbolic equation Au_xx- Bu_yy= 0 has "characteristic curves" \sqrt{A}x+ \sqrt{B}y= 0 and \sqrt{A}x- \sqrt{B}y= 0. Setting s= \sqrt{A}x+\sqrt{B}y and t= \sqrt{A}x- \sqrt{B}y reduces the equation to
4\sqrt{AB}u_{st}= 0 (similar to the way x²- y²= 1 can, by a rotation of axes, by reduced to x'y'= 1) which is easy to solve: dividing by the constants, u_st= 0. Since the derivative of u_s with respect to t is 0, u_s must depend only on s: u_s= f(s) for any function f. Then integrating again, u(s,t)= F(s)+ G(t) where F(s) is an anti-derivative of f and G is the "constant of integration" which, since this is a partial derivative with respect to s, may depend on t.
The General solution to Au_xx- Bu_yy= 0 is
F(\sqrt{A}x+\sqrt{B}y)+ G(\sqrt{A}x- \sqrt{B}y) where F and G can be any twice-differentiable functions of a single variable.

Since elliptic and parabolic equations don't have two separate characteristic curves, they cannot be done that way.

Finally, if you allow variable coefficients: A(x,y)u_xx+ B(x,y)u_yy= something, the equation may be "elliptic" for some values of x and y, "parabolic" or "hyperbolic" for others.

 

[Unskilled]

Thx, i understand a bit now :D:D
I have a question, you said that if i had Auxx-Buyy=something i would have a hyperbolic, is it also true for -Auxx-Buyy=something?
And for parabolic must i be Auxx-Buy=something or can it also be Auyy-Bux=something?
Thx for the help! :)