Partial Differential Equations!

[EvilDarkMagicians]

Says to self, I HAT MATH, then replies, Then why are you in Honors Algebra 2 in 10th Grade?

You hat maths so do I

[UnhealthyTruthseeker]

Code: Select all

"Find x > 3 such that

ln(x) < x^(0.1)"

Bet no one on here can figure this one out

x^(0.1) = ln(x) Is the lower (not included) limit

x = ln(x^10)

e^x = x^10

Ok, this takes Newton's method. I'm not going to use Newton's method to get the upper and lower bounds of this region. Fuck that shit.

[EvilDarkMagicians]

Code: Select all

"Find x > 3 such that

ln(x) < x^(0.1)"

Bet no one on here can figure this one out

x^(0.1) = ln(x) Is the lower (not included) limit

x = ln(x^10)

e^x = x^10

Ok, this takes Newton's method. I'm not going to use Newton's method to get the upper and lower bounds of this region. Fuck that shit.

I thought you were going to do my maths homework for me.
DAMN.
I love maths people they are normally so interesting in real life.

[Taeshan]

Code: Select all

"Find x > 3 such that

ln(x) < x^(0.1)"

Bet no one on here can figure this one out

x^(0.1) = ln(x) Is the lower (not included) limit

x = ln(x^10)

e^x = x^10

Ok, this takes Newton's method. I'm not going to use Newton's method to get the upper and lower bounds of this region. Fuck that shit.

I thought you were going to do my maths homework for me.
DAMN.
I love maths people they are normally so interesting in real life.

funny

[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.

We dove right into the heat equation in Intro to PDE. It's curl-into-a-ball worthy painful math. Even in one dimension, formulating the general solution for the heat equation for an insulated pipe with variable cross sectional area is an enormous pain in the ass.



Where q is heat sourcing.

Whee, thermodynamics. At least my Fluid Mechanics course was condensed enough that the wild effluent heat in a reaction was only interpreted in an overall energy loss value, instead of necessitating its own calculation...

[UnhealthyTruthseeker]

We dove right into the heat equation in Intro to PDE. It's curl-into-a-ball worthy painful math. Even in one dimension, formulating the general solution for the heat equation for an insulated pipe with variable cross sectional area is an enormous pain in the ass.



Where q is heat sourcing.

If you were modeling two surfaces made of different things in contact (assume not external heat sourcing), how would you model thermal diffusivity? It would seem like the diffusivity would be a Heaviside operator of x (we're in just one dimension for now), because you'd have an immediate jump discontinuity right at the contact between surfaces. The equation for X(x) would end up being:

X''(x) + k²(a + b*u(x-x₀))*X(x) = 0

Where k represents the collection of eigenvalues for the system, a is the thermal diffusivity of surface one, a + b is the diffusivity of surface two, u is the unit step function or Heaviside operator, and x₀ is the x coordinate at which the surfaces contact.

How would one solve this? Laplace transforms wouldn't work. Would some sort of discontinuous Taylor series work?

[Deus Malum]

We dove right into the heat equation in Intro to PDE. It's curl-into-a-ball worthy painful math. Even in one dimension, formulating the general solution for the heat equation for an insulated pipe with variable cross sectional area is an enormous pain in the ass.



Where q is heat sourcing.

If you were modeling two surfaces made of different things in contact (assume not external heat sourcing), how would you model thermal diffusivity? It would seem like the diffusivity would be a Heaviside operator of x (we're in just one dimension for now), because you'd have an immediate jump discontinuity right at the contact between surfaces. The equation for X(x) would end up being:

X''(x) + k²(a + b*u(x-x₀))*X(x) = 0

Where k represents the collection of eigenvalues for the system, a is the thermal diffusivity of surface one, a + b is the diffusivity of surface two, u is the unit step function or Heaviside operator, and x₀ is the x coordinate at which the surfaces contact.

How would one solve this? Laplace transforms wouldn't work. Would some sort of discontinuous Taylor series work?

Brain Asplode.

We haven't gotten that far.

[UnhealthyTruthseeker]

Whee, thermodynamics. At least my Fluid Mechanics course was condensed enough that the wild effluent heat in a reaction was only interpreted in an overall energy loss value, instead of necessitating its own calculation...

It is kinda nice that most of the equations that nature obeys are linear and separable, though, isn't it? The only really issues with non-linearity and separability that tend to occur in most PDE's are whenever one has the potential for shear stress. (Navier-Stokes, General Relativity, and rigid body mechanics come to mind.) Of course, a whole bunch of ODE's are non-linear. (Think about the pendulum equation.)

[UnhealthyTruthseeker]

Brain Asplode.

We haven't gotten that far.

I was trying to think of the 2^nd integral of the unit step function, and I came up with x²*u(x-x₀). I think that this will somehow figure into the solution, but I've yet to find the chance to test it. The whole jump discontinuity in thermal diffusivity that happens at the point of contact between surfaces is a bitch.

[PainsOfGod]

It was 1 half of alberts mind.

[UnhealthyTruthseeker]

It was 1 half of alberts mind.

Wat?

[UnhealthyTruthseeker]

Brain Asplode.

We haven't gotten that far.

Actually, I just figured it out. Prior to x₀, it has the behavior of A*sin(sqrt(a)*kx) + B*cos(sqrt(a)*kx). Once you are at x₀, its behavior abruptly changes to C*sin(sqrt(a + b)*kx) + D*cos(sqrt(a + b)*kx). One boundary condition that you should subject this to is that the limit as x -> x₀ is the same in both directions. As A*sin(x) + B*cos(x) allows for an arbitrary phase shift, this boundary condition can be met.

Yay!

[UnhealthyTruthseeker]

Now that I think about it though, it would make more (physical not mathematical) sense to put the a + u(x - x₀) on the d/dt term. This would manifest as a time dependence difference in the two rods that would occur as an instantaneous jump discontinuity at the point of contact.

[UnhealthyTruthseeker]

For more complex thermal diffusivity functions, you figure out the hypersurfaces of constant k and solve a separate diff eq. for each. These hypersurfaces are somewhat like the characteristics of this PDE.

[Christmahanikwanzikah]

Whee, thermodynamics. At least my Fluid Mechanics course was condensed enough that the wild effluent heat in a reaction was only interpreted in an overall energy loss value, instead of necessitating its own calculation...

It is kinda nice that most of the equations that nature obeys are linear and separable, though, isn't it? The only really issues with non-linearity and separability that tend to occur in most PDE's are whenever one has the potential for shear stress. (Navier-Stokes, General Relativity, and rigid body mechanics come to mind.) Of course, a whole bunch of ODE's are non-linear. (Think about the pendulum equation.)

Wooo, Navier-Stokes! At least we were short on time and couldn't touch on del squared operators all too much...

That, and I still have the book and notes from class. XD

[UnhealthyTruthseeker]

Wooo, Navier-Stokes! At least we were short on time and couldn't touch on del squared operators all too much...

That, and I still have the book and notes from class. XD

Laplacian functions are great and appear everywhere in physics. You shouldn't be saying "At least we didn't have time for this." You should be saying "It's a shame we didn't have time for del squared operators."

[UnhealthyTruthseeker]

I've just come to the opposite conclusion of Deus. The heat equation is awesome.

[Christmahanikwanzikah]

Wooo, Navier-Stokes! At least we were short on time and couldn't touch on del squared operators all too much...

That, and I still have the book and notes from class. XD

Laplacian functions are great and appear everywhere in physics. You shouldn't be saying "At least we didn't have time for this." You should be saying "It's a shame we didn't have time for del squared operators."

Considering how tough the class was, I'm happy enough to have passed, heh. If the course wasn't packed into 10 weeks, then I'd be more appreciative.

[UnhealthyTruthseeker]

Considering how tough the class was, I'm happy enough to have passed, heh. If the course wasn't packed into 10 weeks, then I'd be more appreciative.

PDE's r0xx0r!

[New Genoa]

A constant function and e^x are walking on Broadway. Then suddenly the constant function sees a differential operator approaching and runs away. So e^x follows him and asks why the hurry. "Well, you see, there's this differential operator coming this way, and when we meet, he'll differentiate me and nothing will be left of me...!" "Ah," says e^x, "he won't bother ME, I'm e to the x!" and he walks on. Of course he meets the differential operator after a short distance.

e^x: "Hi, I'm e^x"

diff.op.: "Hi, I'm d/dy"

[UnhealthyTruthseeker]

A constant function and e^x are walking on Broadway. Then suddenly the constant function sees a differential operator approaching and runs away. So e^x follows him and asks why the hurry. "Well, you see, there's this differential operator coming this way, and when we meet, he'll differentiate me and nothing will be left of me...!" "Ah," says e^x, "he won't bother ME, I'm e to the x!" and he walks on. Of course he meets the differential operator after a short distance.

e^x: "Hi, I'm e^x"

diff.op.: "Hi, I'm d/dy"

e^x just got fuckin' pwd, bitch!

[Niur]

There is a reason nobody has ever made an NSG topic about higher mathmatics.

[UnhealthyTruthseeker]

There is a reason nobody has ever made an NSG topic about higher mathmatics.

And that reason is?

[New Kereptica]

There is a reason nobody has ever made an NSG topic about higher mathmatics.

And that reason is?

Ignorance breeds contempt.

[Tunizcha]

Really, he should simply be exonerated for trying to help increase your intelligence.