The basic idea of an analog computer is you use op amps to sum voltages. Integration is easy, by adding a capacitor. By hooking up op amps, capacitors, and resistors (usually with a plugboard), you can quickly and easily solve differential equations. (Surprisingly, multiplication by a non-constant value is hard with an analog computer.)
The nice thing about analog computers is they gave the answer almost instantaneously. You could turn the dials and experiment with different parameters interactively. A digital computer in 1970, on the other hand, might take minutes to solve the same equation. This also made analog computers useful for real-time applications such as flight simulators.
A big problem with analog computers is that the accuracy is limited by your components. If you want 0.1% accuracy, you need expensive 0.1% resistors and capacitors. With a digital computer, you can get as many bits of accuracy as you want using cheap components. Basically, Moore's law killed off analog computers.
That being said, that exam was substantially shorter and easier than the Cal II exam I had to take!
Researching this topic also qualifies you to write technobabble for sci-fi movies. Like "operational trans impedance amplifier."
It's simply criminal how underutilised the technique of Automatic Differentiation is! It can solve incredibly complex differentials to machine precision, good numerical stability, and minimal tuning or hand-holding.
Lots of these "difficult" problems aren't, it's just that people aren't aware of the solution because it's not commonly taught in a University setting. Automatic Differentiation is one of those techniques that's "boring" and "numerical", so there are no courses that teach it.
An awful lot of tertiary education has been caught up in the publish-or-perish mentality. Unfortunately for AD, it is not "impressive". Differentiating hideously difficult expressions symbolically is impressive. Publishing a paper packed full of enormous, scary looking expressions is impressive. Merely saying that "we used AD and it worked just fine" is unimpressive.
For decades now we've been incentivizing intellectual wankery over practical utility.
http://www.newsteelconstruction.com/wp/wp-content/uploads/Te...
Besides, the set of problems you can solve is different from the set of problems you can solve quickly. When discussing analog and digital computers, the speed difference isn't all that relevant, but you seem to be confusing them here. Quantum computers do not solve any problem you can't solve on a classical computer, they just solve some faster.
What I'm saying is that even quantum annealers aren't really faster at any practical problem than a comparable supercalculator as of yet.
A much simpler example where naive numerical methods do not work, is any first-order equation with converging characteristics. For example the inviscid burgers equation u_t + u u_x = 0. Even if you start your evolution with a smooth profile, it will form discontinuities. When you try to discretize it, computing derivatives by finite differences does not work at all, and the simplest upwind schemes have a lot of diffusion (which does not correspond to the model).
Probably just a typo, but it's not hard to find particular solutions. The question is more whether solutions exist for all reasonable initial conditions. Here's a famous particular solution:
https://en.wikipedia.org/wiki/Taylor%E2%80%93Green_vortex
I've used a variant of this one to test CFD software.
The Millenium prize problem has a succinct statement of what's under question:
> In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.
https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...
Keep in mind that this is for incompressible Navier-Stokes with a constant viscosity. For inviscid flows you could get discontinuities as you've indicated with the inviscid Burgers equations, and I know that existence and uniqueness problem for the incompressible Navier-Stokes equations has been proved for certain uncommon viscosity laws (which should be reasonably realistic as a constant viscosity is not quite right).
(And of course there are other complications not considered in this problem like boundary conditions.)
Indeed, a fluid eternally at rest is surely a solution :)
Monte Carlo methods are used to solve partial differential equations, I found both approaches similar in the sense that they look at how the system actually/could behave, instead of looking at the equations solely.
You're not really simulating every outcome, just a "large enough" representative sample.
As an aside, my dad was a chemical engineer and Newton-Raphson was his equivalent everything-is-a-nail numerical method. We used to use brent root and monte carlo simulation in the same way.
Numerical methods are awesome in general.
https://en.m.wikipedia.org/wiki/Field-programmable_analog_ar...
I always thought that that way of thinking was pretty neat. Can the same abstraction have a different physical manifestation that's cheap to create? If you kinda squint a bit, it's what we do with computers.