What are the Different Types of Poisson's Equation?

In summary, if a, b, c, d, e, f, and g are constants, an elliptic type equation will be produced if b^2-4ac<0.
  • #1
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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.
 
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  • #2
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
Auxx+ Buyy= something is "elliptic", A and B are numbers of the same sign. Any equation of the form Auxx- Buyy= something is "hyperbolic". Any equation of the form Auxx- Buy= 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 Auxx- Buyy= 0 has "characteristic curves" [itex]\sqrt{A}x+ \sqrt{B}y= 0[/itex] and [itex]\sqrt{A}x- \sqrt{B}y= 0[/itex]. Setting [itex]s= \sqrt{A}x+\sqrt{B}y[/itex] and [itex]t= \sqrt{A}x- \sqrt{B}y[/itex] reduces the equation to
[itex]4\sqrt{AB}u_{st}= 0[/itex] (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, ust= 0. Since the derivative of us with respect to t is 0, us must depend only on s: us= 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 Auxx- Buyy= 0 is
[itex]F(\sqrt{A}x+\sqrt{B}y)+ G(\sqrt{A}x- \sqrt{B}y)[/itex] 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)uxx+ B(x,y)uyy= something, the equation may be "elliptic" for some values of x and y, "parabolic" or "hyperbolic" for others.
 
  • #3
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! :)
 

1. What is a Poisson equation?

A Poisson equation is a type of partial differential equation that relates the second derivative of a function to the function itself. It is commonly used in mathematics and physics to model physical phenomena such as heat transfer, electrostatics, and fluid dynamics.

2. What is the difference between a homogeneous and inhomogeneous Poisson equation?

In a homogeneous Poisson equation, the function being solved for is equal to 0, while in an inhomogeneous Poisson equation, the function is equal to a non-zero constant. This constant is often a source term that represents the influence of external forces or boundary conditions on the system being modeled.

3. How is a Poisson equation solved?

A Poisson equation is typically solved using numerical methods such as finite difference, finite element, or boundary element methods. These methods discretize the equation and solve it iteratively to approximate the solution. Alternatively, analytical solutions may be possible for simpler or special cases of the equation.

4. What are some applications of the Poisson equation?

The Poisson equation has numerous applications in various fields, such as electrostatics, fluid mechanics, heat transfer, and image processing. It is also used in the study of diffusion processes, population dynamics, and financial mathematics.

5. How does the Poisson equation relate to Laplace's equation?

The Poisson equation is a generalization of Laplace's equation, which is a special case where the function being solved for is equal to 0. Both equations are used to model physical phenomena, but the Poisson equation also takes into account external sources or boundary conditions, making it more versatile in its applications.

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