Question

- Partial Differentiation: Heat Equation

Answer 1

PHOTO of question.

Answer 2

i had prepared a response to this and in the process of checking my input i stumbled across this...
http://tutorial.math.lamar.edu/Classes/DE/SolvingHeatEquation.aspx
...which takes it head on, and with background and some "eigen stuff" that i hadn't really thought about.

this is something that is on my bucket list too, so please feel free to come back with observations/questions/etc.

@SithsAndGiggles
@freckles
@Empty

Answer 3

http://tutorial.math.lamar.edu/Classes/DE/SeparationofVariables.aspx there is also seperation of variables

Answer 4

Interestingly f(x) can be pretty much ANY function you like. What about the boundary conditions? Notice even here they have

\[u(0,t)=u(L,t)=0\]and then have \[u(x,0)=40\]
Clearly at x=0 and t=0 you can't have \[u(0,0)=0=40\] So what gives?

Forget the math for a second and let's actually look at what's happening here. Really we're just looking at basically a metal rod of length L. This metal rod can have any temperature distribution on it that we like, except for maybe infinite, that might be a little too unphysical, and the temperature at a point x on the rod is given by f(x), or really at the initial time u(x,0).

Now what are we doing to this rod? Well we're hooking up some constant temperature, 0 degrees cooling right at the ends of the rod and then observing what happens after we hook these up.

So this problem itself feels incredibly arbitrary, your temperature distribution doesn't have to really look like sines or cosines, in fact your physical problem is something that is completely independent of the representation of it. However, if you can represent your temperature distribution as a Fourier series, we see that each individual contribution of the temperature decays in a straightforward way, it exponentially decays dependent on the square of the frequency... Which is kind of weird but also something we can reason out by just looking at the original differential equation.

Yeah, so pretty cool and difficult to get comfortable with, but really all of this boils down to linear algebra, perhaps you are familiar with Taylor series, representing a function as an infinite sum of polynomials? A Fourier series is just like this. At the end of the day, both form a complete basis for representing most arbitrary functions, and these decompositions allow us to do fancy things to them.