Dissipative and dispersive derivatives

[feynman1]

Why are even order derivatives dissipative and odd order derivatives dispersive?

 

[jasonRF]

Who says they are? What is the context here?

 

[jim mcnamara]

Please let us know where you found this. Book, textbook, online URL, etc. We cannot do much with what we have so far.

 

[feynman1]

Who says they are? What is the context here?

Sorry I forgot the context or source. This notion has been with me for 10 years. This at least applies to PDEs, not sure about ODEs. Simplest examples, traveling wave equation, heat equation.

 

[jasonRF]

Consider
$$ c_1 \, \frac{\partial u_1}{\partial x} + \frac{\partial u_1}{\partial t} = 0 $$
which has solutions of the form $u_1 = f_1(x - c_1 t)$, and
$$ c_2^2 \, \frac{\partial^2 u_2}{\partial x^2} - \frac{\partial^2 u_2}{\partial t^2} = 0 $$
which has solutions of the form $u_2 = f_2(x - c_2 t) + g_2(x + c_2 t)$.

So one admits traveling waves only in one direction, and admits them in both directions. Neither has any dissipation. I would say that neither has dispersion, either, since the speed of propagation is not dependent on frequency or wavelength. Perhaps you are using a different definition of dispersion than I am? In any case, they have the same types of solutions even though one has only second order derivatives and one has only first order derivatives.

Unless I'm misunderstanding, your 'notion' is false.

jason

 

[S.G. Janssens]

I also had to look this up. It's better to be a bit more specific, otherwise one risks a "guess thread".

As far as I came to understand, this has less to do with the PDE, but more with its discretization. For example, chapter 7 in J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods is dedicated to this. (It seems to be discussed in most textbooks on the topic.)

 

[feynman1]

Consider
$$ c_1 \, \frac{\partial u_1}{\partial x} + \frac{\partial u_1}{\partial t} = 0 $$
which has solutions of the form $u_1 = f_1(x - c_1 t)$, and
$$ c_2^2 \, \frac{\partial^2 u_2}{\partial x^2} - \frac{\partial^2 u_2}{\partial t^2} = 0 $$
which has solutions of the form $u_2 = f_2(x - c_2 t) + g_2(x + c_2 t)$.

So one admits traveling waves only in one direction, and admits them in both directions. Neither has any dissipation. I would say that neither has dispersion, either, since the speed of propagation is not dependent on frequency or wavelength. Perhaps you are using a different definition of dispersion than I am? In any case, they have the same types of solutions even though one has only second order derivatives and one has only first order derivatives.

Unless I'm misunderstanding, your 'notion' is false.

jason

Then perhaps it applies to PDEs with only d/dt on the LHS. I define dispersion as going out of phase, so as long as some wave is translated, the 'phase' changes hence dispersion comes. Am I wrong?

 

[jasonRF]

Then perhaps it applies to PDEs with only d/dt on the LHS.

If you only allow first order time derivatives, do you at least allow systems of PDEs? If not then this is a highly restrictive class of equations.

I define dispersion as going out of phase, so as long as some wave is translated, the 'phase' changes hence dispersion comes. Am I wrong?

What do you mean when you write ‘going out of phase’? Could you give us an example, including equations, so we can understand what you mean?

jason

 

[feynman1]

If you only allow first order time derivatives, do you at least allow systems of PDEs? If not then this is a highly restrictive class of equations.
What do you mean when you write ‘going out of phase’? Could you give us an example, including equations, so we can understand what you mean?

jason

No idea about systems of them. E.g. ut=ux+uxx with I.C. u(t=0)=dirac(x=0). Then the peak will decay due to uxx and will translate along x due to ux (out of phase with x=0 being the peak).

 

[jasonRF]

That is a very limited class of pdes. It also seems like by 'dispersive' you simply mean disturbances propagate in a wave-like manner.

In any case, to understand why the different derivatives do what they do, just think about what the spatial derivatives physically mean: first derivative is spatial slope, and second derivative is concavity. To help your thinking you can use the simple approximation $$u(x,t+\delta t) \approx u(x,t) + \delta t \left[ u_x(x, t) + u_{xx}(x,t) \right] $$, and then pick $\delta t = 1$.

Even just looking at the signs of the two terms can be instructive. For example, where the curve is concave up the second derivate will act to increase the value of $u$, and where it is concave down it will decrease the value of $u$. So the second derivative acts to fill-in valleys and lower hills. What would you say the first derivative does?

I would actually look at the two equations $v_t = v_x$ and $w_t = w_{xx}$ separately. On the same axis sketch $v(x,t=0)$, $v_x(x,t=0)$ and $v(x,t=1)$. Make a separate axis to sketch $w(x,t=0)$, $w_{xx}(x,t=0)$ and $w(x,t=1)$.

This kind of simple exercise should help you understand what is going on.

 

[feynman1]

$$u(x,t+\delta t) \approx u(x,t) + \delta t \left[ u_x(x, t) + u_{xx}(x,t) \right] $$, and then pick $\delta t = 1$.

Thanks but how did you get this equation?

 

[jasonRF]

$u_t(x,t) \approx \frac{u(x,t+\delta t) - u(x,t)}{\delta t}$.

 

[feynman1]

$u_t(x,t) \approx \frac{u(x,t+\delta t) - u(x,t)}{\delta t}$.

Oh you put ut=ux+uxx in. I agree with what you said, but what's wrong with my post 9 or earlier?

 

[jasonRF]

Oh you put ut=ux+uxx in. I agree with what you said, but what's wrong with my post 9 or earlier?

There is nothing wrong with post 9. But you were the one who asked the question about why the derivatives effect the solution the way they do, and I was simply trying to lead you on a path of answering your own question.

 

[feynman1]

There is nothing wrong with post 9. But you were the one who asked the question about why the derivatives effect the solution the way they do, and I was simply trying to lead you on a path of answering your own question.

I understand 1st and 2nd derivatives, but could you explain about higher order ones?

 

[jasonRF]

Have you sketched what the derivatives look like for example functions?

 

[feynman1]

Have you sketched what the derivatives look like for example functions?

yes they are understood by undergrads

 

[jasonRF]

yes they are understood by undergrads

Agreed! But you haven’t exactly given us any evidence that you have done any work at all on your own to support your notion. How about you explain to us why you have this notion?

 

[Vrangr]

Why are even order derivatives dissipative and odd order derivatives dispersive?

With the heat equation, the change of a function at a point is proportional to the curvature of the function at that point. The function changes in such a way to minimize the curvature which is why it flattens out and disperses (I'd recommend 3Blue1Brown's video on differential equations and the heat equation). The dissipation of say air resistance comes from the first derivative. It says that an object in motion will be pushed in the direction opposite it's motion and that the force is proportional to the velocity. So it slows rapidly and stops. I think specific examples can be explained but no strict rule is to be found. Hope this helps.

Source https://www.physicsforums.com/threads/dissipative-and-dispersive-derivatives.1006592/

 

[Mark44]

Source https://www.physicsforums.com/threads/dissipative-and-dispersive-derivatives.1006592/

This link is unhelpful, as it points to the same thread that it is in.