We take our computers so much for granted these days that we usually don’t even stop to ask ourselves how they work. But no more than a little over half a century ago, digital computers did not exist. How were complex and tedious computations to be carried out? Obviously computing everything by hand is an option - but this is error prone and probably takes far too long. The other option is to construct a computer to do it for you.

But I thought you said computers back then didn’t exist!

While digital computers are a fairly new invention, analog computers have been around for centuries, and in fact there is a long history of computers to integrate and solve ODEs. Perhaps one of the greatest reasons for this is that much of the physical phenomena in our world is measured by calculus - so if we can set up some sort of simulation quick and easily, we can do complex numerical mathematics with it.

According to Wikipedia, the Antikythera mechanism (an ancient Greek mechanical calculator used in their Astronomy) is believed to be the first analog computer, dating back to 100 B.C. That might be a little too primitive. Let’s fast forward about two millenniums. First conceptualized by James Thomson in 1876, the Differential Analyser allowed for an automated solving of differential equations mechanically.

But it’s extremely large, and there’s something far more interesting that can be done to perform numerical calculus: the electric analog computer.

Granted, the electric analog computer died out pretty quickly because of its coincidence with the emergence of the digital computer, but it’s still really interesting, especially since you can create one in your own home!

Several simple circuits form the building blocks of this electric calculus, all of which require the use of operational-amplifiers.

First, you have the integrator:

If you want certain functions to be integrated, you just choose different waveform functions V(t).

As for differentiation:

So how do you actually solve a differential equation with one of these things?

There is no general solution, as each differential equation requires its own unique circuit. However, one could in theory create a series of switches that would rewire the circuit to different arrangements depending on the problem at hand. An example of a solver for a particular differential equation can be seen here:

x” + ax’ + bx = 0

is solved by:

For the values of the unspecified capacitors and resistors, these depend on the particular constants in the problem at hand.

Of course, these functions aren’t perfect, as would be hinted by their early demise. They are very susceptible to noise, giving rougher approximations than their digital counterparts, and in some cases (although not all) were slower when being used to solve differential equations.  Still, they are a great intellectual curiosity, and stress that the medium by which we compute is just as important as the computation itself.

Sources:

http://www.owlnet.rice.edu/~elec301/Projects99/anlgcomp/

http://technology.open.ac.uk/tel/people/bissell/bletchley_paper.pdf

http://en.wikipedia.org/wiki/Differential_analyser

http://www.allaboutcircuits.com/vol_3/chpt_8/11.html

physics.umbc.edu/~mcmillan/PHYSICS340/Labs2008/Lab8.doc