Solving Cauchy PDEs using the Method of Characteristics

[mmmboh]

Could someone tell me where to start? I tried separating variables, which got me no where (plus we haven`t technically learned it), and I tried putting it into a form of D^2U, but I couldn`t figure that out either. Please help.

Thank you.

 

[mmmboh]

Anyone?

 

[hunt_mat]

Have you tried Fourier transforms?

 

[mmmboh]

We haven`t learned that. There must be a more simple way. :S

 

[hunt_mat]

What HAVE you done on your course? Once I know this I can select the appropriate technique.

 

[mmmboh]

Cauchy problems for the heat and wave equations. Poisson equation, laplace equation, characteristic curves, dirichlet problem, finite difference method, advection in 1d..

 

[hunt_mat]

Okay have you done characteristics for 3 variables rather than two?

The way to do this in 1d is to use characteristic co-ordinates.

Mat

 

[mmmboh]

No we`ve only done it for two variables :S...I`ll give it a shot, but can you let me know what substitutions to use please? Do i just introduce a third variable equal to like y-ct or what :S

 

[hunt_mat]

Hmmm, come to thin about it have you done anything in 2 spatial dimensions? Can you extend what you did on the cauchy problem for the wave eqution?

 

[mmmboh]

We looked at the Cauchy problem in 1 spatial dimension. Most of what we have done is in two dimensions total.

 

[hunt_mat]

I think the question is saying that it is radially symmetric, do you agree?

 

[mmmboh]

Yes that`s what it seems like.

 

[hunt_mat]

So what is the radially symmetric part of Laplace operator $$\nabla^{2}$$

 

[mmmboh]

In polar coordinates, it is (1/r)(d/dr)(rdu/dr)...Am I suppose to take the laplacian of u, and see something?

 

[hunt_mat]

Expand that out and denote v=ru, and what equation do you have now?

 

[mmmboh]

But u(x,y,t) is not in terms of r, can I write u(r,t) instead and use the laplacian on that? If that`s what you mean..or do you mean something else?

I can expand it, and get (1/r)(du/dr)+(d/dr)(du/dr), but there is no u*r in this.

If I let u=v/r, I get v/r^3.

 

[hunt_mat]

You should end up with the 1D equation for v, solve this in the usual manner and the problem is easily solved.

 

[mmmboh]

I get v/r^3 = 0 :S...I edited my post if you didn`t see, can you check if that`s what I`m suppose to get? (but I still don`t yet see how this will show that u vanishes for the given condition, or why we can assume u satisfies laplace`s equation).

Thanks for the help btw.

 

[hunt_mat]

Just to confirm the equation you should be solving is:

$$\frac{\partial^{2}u}{\partial t^{2}}-c^{2}\left(\frac{\partial^{2}u}{\partial r^{2}}+\frac{2}{r}\frac{\partial u}{\partial r}\right) =0$$

Use $$v(t,r)=ru(r,t)$$ to obtain a 1D wave equation, turn that intop characteristic co-ordinates in the usual way and that will give you your solution.

I am off to bed now.

 

[mmmboh]

So do you mean I should I write u=v/r, and insert that into the equation you wrote? Good night, thanks.

If so, I get v_tt=0, hope that is right!

 

[hunt_mat]

Yes, have you not been doing that?

 

[mmmboh]

I wasn`t before :S, but now I get I get v_tt=0, so I guess I can just integrate that and substitute u in.

 

[hunt_mat]

as u=u(t,r) so is v=v(t,r)

 

[mmmboh]

I get u(r,t)=a(r)*r*t+r*b(r), where a,b are some functions of r. I don`t see how this means u vanished for the given condition though :S

Edit: Oops I didn`t differentiate the v.

 

[hunt_mat]

Calculate:

$$\frac{\partial^{2}v}{\partial r^{2}}$$

to get the folowing equation:

$$\frac{\partial^{2}v}{\partial t^{2}}-c^{2}\frac{\partial^{2}v}{\partial r^{2}}=0$$

Then introduce the co-ordinates:

$$\begin{array}{ccc} \alpha & = & r+ct \\ \beta & = & r-ct \end{array}$$

use the condition you have and this should give you the result you need.