method of characteristics

gtfitzpatrick

1. The problem statement, all variables and given/known data

[itex] x \frac{ \partial u}{ \partial x} + y \frac{ \partial u}{ \partial y}= -x^2u^2[/itex]

2. Relevant equations



3. The attempt at a solution

characteristics are given by
[itex] \frac{ dy}{ dx} = \frac{ y}{ x} [/itex] (a)

and

[itex] \frac{ du}{ dx} = -\frac{x^2u^2 }{ x} [/itex] (b)

So i integrate both equations

but for (a) do i bring the y across which ends up giving ln(y) = ln(x) + k
or leave it where it is and i get y = -yln(x) + k

??

hunt_mat

Hello, still doing characteristics? Why not write the characteristic equations as:
[tex]
\dot{x}=x,\quad\dot{y}=y,\quad\dot{u}=-x^{2}u^{2}
[/tex]
You won't be able to write down a complete solution as you have no initial condition to work from. For your equation for characteristics, your first answer was correct, the characteristics are given by [itex]y=kx[/itex] for k constant.

nickalh

To expand on hunt_mat's answer:
Don't we need to separate the variables before integrating?

1/y dy = 1/x dx


Hmmm, then again you're using partial deriv. notation, which I've only studied Calc. for, not yet Differential Equations.

gtfitzpatrick

thanks for the replies lads. I guess i never total got to grips with methods of chars.!!!
I also have initial conditions of u(x,1) = x for -infinity<x<infinity.

y=kx becomes k=y/x

then from (b) we have [itex] \frac{ du}{ dx} = -\frac{x^2u^2 }{ x}[/itex] which when differentiated gives [itex] u = \frac{ 2}{ x^2} + F(k) [/itex]

and transfer in our earlier value of k gives [itex] u = \frac{ 2}{ x^2} + F(\frac{ y}{ x}) [/itex]
im getting confused now i think in different methods?

hunt_mat

There are essentially two ways for the method of characteristics (I recognise your name from a number of MOC posts, did I post my notes on the subject?)
From one of your calculations you have:
[tex]
\frac{du}{dx}=-xu^{2}
[/tex]
Integrating this equation shows that:
[tex]
\frac{1}{u}=\frac{x^{2}}{2}+F(\xi )
[/tex]
We now paramatrise the initial data, so take [itex](\xi ,1)[/itex] as the point which the characteristic passes through, this will give the initial values as [itex]u(\xi ,1)=\xi[/itex], evaluating the characteristic at this point yields [itex]1=k\xi[/itex], giving a value for k which can then be inserted back into the equation for the characteristic. We now evaluate [itex]u[/itex] at the point [itex](\xi ,1)[/itex] to obtain
[tex]
\frac{1}{\xi}=\frac{\xi^{2}}{2}+F(\xi )
[/tex]
From this you can compute [itex]F(\xi )[/itex] and then from there you can substitute for [itex]\xi[/itex] by using the equation of the characteristic.
Now to find u you have the solution