Partial Differential Equations!

[UnhealthyTruthseeker]

Which one of Schrodinger's equations? Wave equations?

Other than the QM one, what other one's are there?

[Buffett and Colbert]

*head asploads*

[Tunizcha]

*head asploads*

*tapes Buffy's head back together*

[Buffett and Colbert]

*head asploads*

*tapes Buffy's head back together*

*looks at particle diffusion equithingies and melts*

[Christmahanikwanzikah]

Let's discuss Partial Differential Equations!

Laplace's equation is nice to work with because it is homogeneous and linear and hence always separable. It's solutions are many, but they are always of a polynomial or exponential nature. Physically, Laplace's equation usually implies some superposition of wave-like behavior in a scalar field.

The equation for a solenoidal vector field (that the vector field has zero divergence) is also very easy to solve and is always separable, but it also doesn't reveal the entire picture. It cannot tell you how the x component of the vector field depends on an additive term containing the y and z variables, for example. In order to figure out the full behavior of the vector field, you must know its curl field as well.

EDIT: I should correct myself. The solutions to Laplace's equation are always of a polynomial or exponential nature for coordinates in a holonomic basis.

If you take a Fourier transformation of a wave function and that wave function happens to be a sound recording of a human voice you can use it as the first step in creating speech recognition software.

Do you think a Laplace transformation could be equally useful? I've heard that Laplace transformations can have an effect of "noise reduction" I've never actually used them for that though.

The Transform That Dare Not Speak Its Name! That's fighting talk round these parts, sunshine...

I thought I was good at maths til I studied astrphysics at uni. I then discovered that I was, in fact, shit. Thus my attaining only a Desmond. Well, perhaps being a lazy sod also contributed.

As for modelling piss flow, surely chaos theory has to come into play? I mean, there's no rhyme or reason to the act.

It's impossible to even mathematically figure out flow patterns for turbulent flow, so we don't really try to figure it out... we just find things like flow rate and friction and stuff through transport theorems and lots of experimentation. Hence Reynold's Number and the Moody Chart:



Laminar flow, on the other hand, is semi-easily representable through math.

[UnhealthyTruthseeker]

It's impossible to even mathematically figure out flow patterns for turbulent flow, so we don't really try to figure it out... we just find things like flow rate and friction and stuff through transport theorems and lots of experimentation. Hence Reynold's Number and the Moody Chart:



Laminar flow, on the other hand, is semi-easily representable through math.

You can, however, figure out the flows using computational methods, like finite difference methods. You could also use a horrifying technique like a method of successive approximations.

[Natapoc]

It's impossible to even mathematically figure out flow patterns for turbulent flow, so we don't really try to figure it out... we just find things like flow rate and friction and stuff through transport theorems and lots of experimentation. Hence Reynold's Number and the Moody Chart:

Laminar flow, on the other hand, is semi-easily representable through math.

You can, however, figure out the flows using computational methods, like finite difference methods. You could also use a horrifying technique like a method of successive approximations.

Hurry! Someone write a paper on a new method of figuring out turbulent flow patterns mathematically and get a free PHD for your efforts

[UnhealthyTruthseeker]

Hurry! Someone write a paper on a new method of figuring out turbulent flow patterns mathematically and get a free PHD for your efforts

Honestly, you can figure out pretty much any differential equation in a mathematical way, it's just that many of these equations don't have any "nice" solutions, and the method for getting such non-closed form solutions could involve an obnoxious process that technically involves an infinite number of steps, like successive approximations or series solutions!

[Natapoc]

Hurry! Someone write a paper on a new method of figuring out turbulent flow patterns mathematically and get a free PHD for your efforts

Honestly, you can figure out pretty much any differential equation in a mathematical way, it's just that many of these equations don't have any "nice" solutions, and the method for getting such non-closed form solutions could involve an obnoxious process that technically involves an infinite number of steps, like successive approximations or series solutions!

They don't have any known nice solutions Unless there is a proof that a "nice" solution cannot exist there could be an elegant method just waiting to be discovered.

[UnhealthyTruthseeker]

They don't have any known nice solutions Unless there is a proof that a "nice" solution cannot exist there could be an elegant method just waiting to be discovered.

Well, if the equation, though nasty, has some basic properties but is still very nasty, certain theorems guarantee that there are no more solution sets than the nasty-ass sets that you've already attained.

[Risottia]

It's impossible to even mathematically figure out flow patterns for turbulent flow, so we don't really try to figure it out... we just find things like flow rate and friction and stuff through transport theorems and lots of experimentation. Hence Reynold's Number and the Moody Chart:
(snip big pic)
Laminar flow, on the other hand, is semi-easily representable through math.

AIEEE!!! That's... that's... ENGINEERING!

[UnhealthyTruthseeker]

AIEEE!!! That's... that's... ENGINEERING!

Would you prefer having to deal with an incredibly obnoxious solution that isn't closed-form?

[UnhealthyTruthseeker]

Here's another set of nice equations:

Let there be light:





The equation for B_x(x), B_y(y), and B_z(z) is always separable and thus solvable. (basically)

The equation for E_x(x), E_y(y), and E_z(z) is like the Schrodinger equation in that it is separable as long as you can find the right coordinate system.

The equations for things like E(t) and B_x(y,z) are usually not too bad, and are, again, separable if you pick the right set of coordinates.

[Tunizcha]

I still have trouble cranking out solutions for equations nowadays. I understand the theories and the implications of the equations, but not the equations themselves.

[Hydesland]

When I read this thread I feel stupid .

[UnhealthyTruthseeker]

I still have trouble cranking out solutions for equations nowadays. I understand the theories and the implications of the equations, but not the equations themselves.

You won't really understand what curl and divergence mean until you solve a few diff eqs involving them. This is the industry standard undergraduate text for differential equations. It is VERY good and VERY easy to follow.

This book is very good, but it is a graduate school text for PDE's and it assumes that you've already had some undergrad experience with PDE's. If you work this book after taking modern physics I and II, you'll be in way ahead of the curve in grad school.

[Niur]

When I read this thread I feel stupid .

I understand the feeling, however it also makes me feel young.

[UnhealthyTruthseeker]

When I read this thread I feel stupid .

I thought you did something technical.

[Hydesland]

I thought you did something technical.

Eh, I study economics at uni, so when I study calculus, they never bother to apply it to thinks like wave forms and turbulent flows or whatever- it's either just pure equations or it's applied to economic phenomenon. So I don't have any idea what you guys are talking about in most of this thread. Maybe if you did a statistics thread I'd understand more.

[UnhealthyTruthseeker]

Eh, I study economics at uni, so when I study calculus, they never bother to apply it to thinks like wave forms and turbulent flows or whatever- it's either just pure equations or it's applied to economic phenomenon. So I don't have any idea what you guys are talking about in most of this thread. Maybe if you did a statistics thread I'd understand more.

But you do work with differential equations, don't you? (Or are most economics equations uni-variate?)

[UnhealthyTruthseeker]

There's a strong connection between the method of separation of variables for PDE's and multi-variable statistics. Does that help?

[Hydesland]

Eh, I study economics at uni, so when I study calculus, they never bother to apply it to thinks like wave forms and turbulent flows or whatever- it's either just pure equations or it's applied to economic phenomenon. So I don't have any idea what you guys are talking about in most of this thread. Maybe if you did a statistics thread I'd understand more.

But you do work with differential equations, don't you? (Or are most economics equations uni-variate?)

Yeah, I did long before uni too. At uni I do things like calculus of multi-variable functions, using Lagrangians and bordered Hessians, so it's not all univariate.

[Hydesland]

Also it's possible I may have done PDE's at one point, but I think I've simply forgotten it.

[Third Spanish States]

Let's discuss Partial Differential Equations!

Damn, you just reminded me of one of the Gauss formulas in electromagnetism. The one where a gigantic calculus involving integration is "needed" just to give as the result either -1 or 1 *headdesk*. And the fact it's totally irrelevant IRL, because anyone worth his pay should use a specialized calculator to do most of this stuff.

[Tunizcha]

I still have trouble cranking out solutions for equations nowadays. I understand the theories and the implications of the equations, but not the equations themselves.

You won't really understand what curl and divergence mean until you solve a few diff eqs involving them. This is the industry standard undergraduate text for differential equations. It is VERY good and VERY easy to follow.

This book is very good, but it is a graduate school text for PDE's and it assumes that you've already had some undergrad experience with PDE's. If you work this book after taking modern physics I and II, you'll be in way ahead of the curve in grad school.

I'm going to High School next year, so I'll be looking to you as my mentor