Solve 1st Order PDE: $u_y+f(u)u_x=0$

[Markov2]

Solve $u_y+f(u)u_x=0,$ $x\in\mathbb R,$ $y>0,$ $u(x,0)=\phi(x).$

What's the easy way to solve this? Fourier Transform? Laplace Transform?

 

[Mathwebster]

I would say separation of variables.

 

[Ackbach]

I would say separation of variables.

Wouldn't you have to say something about $f$ in order for that to work?

 

[Mathwebster]

Wouldn't you have to say something about $f$ in order for that to work?

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!

 

[Markov2]

Okay, that consists on putting $u(x,y)=h(x)g(y)$ right?

 

[Mathwebster]

I would say separation of variables.

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. (Doh)

 

[Markov2]

Oh, could you show me or where can I learn the method?

 

[Mathwebster]

Given a PDE of the form

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

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.

 

[Markov2]

Well I have the Farlow, the "partial differential equations" and etc, is it good? Could you give examples as you said? Thanks.

 

[Ackbach]

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!

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. (Doh)

Yeah, I was more getting at the "separability" of $f$. Characteristics doesn't seem to have that drawback in this case.

 

[Jose27]

Using the method of characteristics you shoudl arrive at something like \(u(x,t)=\phi(y_0(x,t))\) where \(y_0(x,t)\) is the intersection of the \(x\)-axis with the characteristic passing through \((x,t)\).

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.

 

[Markov2]

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.

Sorry for the delay of the reply, but those books are online? Can you give the links if so?

Thanks!