TITLE: Why aren't all important differential equations solved and codified once and for all?
QUESTION [23 upvotes]: This is not a number question, and I hope it is on topic.
I always wondered how come, with modern supercomputers, the most important DE, especially the ones that refer to Physics, haven't been patiently solved one by one, and codified for reference, the way a century ago they did with sines, trig values and log tables.
Do you have an answer and an explanation, if the answer is negative?

REPLY [4 votes]: There are many possible ways to answer this question, and I recommend perusing all of the answers given.
My own take is this: You are correct that people used to work with tables for sin, log etc. But now we don't, and neither do computers. A computer doesn't store a table of values for a function like sin; rather it computes on the spot any value it needs.
Why do you think that is?
The reason is space-time tradeoff. Sure, having a ready-made table could save us some computation time. But it will take up a lot of space - space which could have been used for something more productive.
If the table is precise enough, it can easily be so large it doesn't fit in memory. You'll have to store it on a hard drive, and access it every time you need to find a value for the function. I/O access can easily take a million times longer than just computing the damn thing, so not only you've wasted space, you end up with slower computation.
For many types of computation, using a lookup table wisely can greatly increase performance without requiring a lot of storage. But storing a table of values for a function like sin is not an example of wise usage.
The problem is orders of magnitude worse for storing numerical solutions to differential equations. You don't have a function of one variable, but of several - you need to store values for every combination of boundary values, coefficients, and the values of actual variables (the equation can be partial and have more than one variables). If you have 5 parameters, a million possibilities per parameter becomes a nonillion entries and millions of yottabytes of data. The parameters aren't necessarily bounded because you don't have neat periodic behavior like sin, the solutions may need to be even more precise, there could be chaotic behavior, and no matter how precise the table is, there will always be a case that you need even more precision for a specific application. For a PDE, the boundary condition isn't even a finite set of numbers but a function, and you'll want a solution for each possibility.
So the necessary conclusion is - if you want a numerical solution for a differential equation, it is much, much easier to simply compute it on the spot with the specific parameters you need.