- The analytical solution gives the exact solution.
- The numerical solution gives the approximate solution.
Consider this function: and imagine you want to know the result of
So according to your calculus course in order to answer this you use the fundamental theorem of calculus you find the primitive and the answer is:
Now imagine the function is ten times more complicated than this one and after hours of trying to solve it you discover that every technique you learned in your calculus course is useless (an example of a function like this is )
You know there is an answer because every continuous function has an integral, so what do you do?
Well, there is where numerical solutions come to use.
Everyone who has taken a proper calculus course before learning how to solve integrals, they learn what is integral. As an introduction you see the following definition:
Calculating this limit is sometimes almost impossible but what if you only want some degree of precision (for example 10 digits), then you can do as many iterations of this formula until you fill fine with your answer (even if it is not an exact solution).
The fist procedure in my answer is an example of an analytical solution and the second is an example of a numerical solution.