Stuck on a characteristic PDE

I can't figure this out. The equation is $\displaystyle u_{xx}+4u_{yy}=0$. Solving for y', I get y1 = 2ix + c1 and y2 = -2ix + c2.

Then v = y1 - 2ix and w = y2 + 2ix.

I keep getting $\displaystyle -4u_{vv}+8u_{vw}-4u_{ww}=0$ and $\displaystyle u_{vv}+2u_{vw}+u_{ww}=0$ from the second partials with respect to x twice and y twice, respectively. This doesn't tell me anything, because these two equations are equivalent. Please advise.

Re: Stuck on a characteristic PDE

Quote:

Originally Posted by

**phys251** I can't figure this out. The equation is $\displaystyle u_{xx}+4u_{yy}=0$. Solving for y', I get y1 = 2ix + c1 and y2 = -2ix + c2.

Then v = y1 - 2ix and w = y2 + 2ix.

I have no idea what you're doing there. "Solving for y' "?? "Then v = ... and w = ..."??

The equation you gave mentions one function only, u(x,y). So what's y'? What are v and w? Why do you have complex numbers - is u complex valued, or do they arise in whatever way you went about solving this?

Remember - we can't read your mind. You'll need to give more details before we can even understand the problem.

Re: Stuck on a characteristic PDE

Here is what I know. I'm just putting down what I can; I just learned this and do not have the hang of it yet.

The given differential equation is $\displaystyle u_{xx}+4u_{yy}=0$.

The form of that equation is $\displaystyle Au_{xx}+2B_{xx}+Cu_{yy}=0$, and since AC-B^2 > 0, it's an elliptic equation.

Then I'm supposed to solve the characteristic equation $\displaystyle Ay^{'2}-2By^{'}+C=0$ I have no idea why; I'm just going through the motions at this point. I am just learning this stuff. Then, after that, I'm supposed to transform x and y into v and w. The book does not give any examples of how to do it; I am going only on my notes.

Please; I really need help with this.

Re: Stuck on a characteristic PDE

It appears that all that you get from PDE is:

$\displaystyle w(\text{x},\text{y})\text{=}c_1 y+2i\text{ }c_2x$

Now what other conditions are there?(Thinking)

Re: Stuck on a characteristic PDE

Thanks for adding a few more details about the problem. It makes a difference.

(PDEs aren't my strong suit, so take what I say with a grain of salt. I had to go review this stuff - so, please, do not take my word for any of this.)

The characteristic equation of a constant coeff 2nd order linear PDE is used to:

1) determine what kind of initial information about the solution allows the problem to be solveable. There are sometimes certain special curves in the x-y plane, characteristic curves, that, when data is provided for them, do *not* permit the PDE to be solved. The characteristic equation finds those curves. However, in the ellipitic case, there are no such bad curves. (Hence there would be no reason for you to bother with them.)

2) help decide on a change of variables that will transform the given PDE into a one in even more canonical form. In the elliptic case, that means a transformation so that the new equation has 2nd order terms like the Laplacian (Laplace's Equation, or Poisson's Equation).

3) Other stuff too, I imagine, that I simply don't know. The method of charateristics uses characteristic curves to try separation of variables, but I don't know if that applies to 2nd order PDEs or not.

There's no "solving" this PDE, as it lacks boundary value conditions. So what is the problem even asking you? There aren't going to be any characteristic curves - so nothing about those is relevant. In your case, I would *guess* that they want you to correctly classify the PDE (it's elliptic) and then transform the PDE into a Lapacian by a change of coordinates, using choices for that transformation as given in your book based on examining the original/charateristic equation. The characteristic equation doesn't have a real solution, but its coefficients do provide a clue how to transform the original PDE into a Laplacian by a change of coordinates.

Sorry - but that's the best I can do given what I know and what you've written. I would suggest rereading the problem carefully to make sure you understand *exactly* what you're being asked to do with that PDE. You're not being asked to solve it (unless you've neglected to tell us about the rest of the problem), because it doesn't have a solution as is other than "all harmonic functions" up to a rescaling.