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- Thread starter Markov
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- #1

- Jan 26, 2012

- 183

I would say separation of variables.

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- #3

- Jan 26, 2012

- 4,192

Wouldn't you have to say something about $f$ in order for that to work?I would say separation of variables.

- Jan 26, 2012

- 183

You mean like some level of smoothness. For sure. In most cases I would use a separation of variables. For special cases of $f(u)$ - something else.Wouldn't you have to say something about $f$ in order for that to work?

Thanks for pointing that out!

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- #5

- Jan 26, 2012

- 183

What I meant to say was the method of characteristics (What was I thinking???) That's what happens when I think mathematics w/o my first cup of tea in the morning.I would say separation of variables.

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- #7

- Jan 26, 2012

- 183

$a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u)$

the method of characteristics requires you to solve the ODEs

$\dfrac{dx}{a(x,y,u)} = \dfrac{dy}{b(x,y,u)} = \dfrac{du}{c(x,y,u)}$

in which you pick in pairs and try and integrate (sometimes it can be tricky as you might have to be clever in how you pick or manipulate the equations)

What book(s) do you have? You could do a google search for examples or I could give you a list of books that have examples.

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- #10

- Jan 26, 2012

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You mean like some level of smoothness. For sure. In most cases I would use a separation of variables. For special cases of $f(u)$ - something else.

Thanks for pointing that out!

Yeah, I was more getting at the "separability" of $f$. Characteristics doesn't seem to have that drawback in this case.What I meant to say was the method of characteristics (What was I thinking???) That's what happens when I think mathematics w/o my first cup of tea in the morning.

For this method I would recommend John's book "Partial differential equations" where he treats in some detail the case \(f(u)=u\). Also, as an aside, putting \( f=g'\) for some \(g\), we can put the equation in the form \(u_t+g(u)_x=0\), and this is known as a scalar conservation law in one dimension; there are books dedicated to these kinds of equations.

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Sorry for the delay of the reply, but those books are online? Can you give the links if so?What book(s) do you have? You could do a google search for examples or I could give you a list of books that have examples.

Thanks!