In Mathematics we have special functions, symbols we know by name, meet quite often, know their form, shape and behaviour. From the simple Sin, Cos, Exp, .. to the Meijer G-Function, which is very general. Some of them are good friends, we can artistically create derivatives, integrals, .. when they dominate in expressions, equations and use them as Ansatz for, say, solving differential equations. New special functions expand the domain of closed form solutions.
Closed form solutions are elegant, but usually they do approximate only small domains. In the vast majority of the cases you need numerical schemes to make the predictions and simulate your models.
To model, say, the material flows, chemical reactions, and conservation of energy in, say, a blast furnace you need to solve systems of dozens of coupled nonlinear PDEs. They need to be solved numerically.
Mid 90ies, I entered into partnership with MathConsult specialized in the sw development of advanced numerical schemes. In 1996 we were lucky being asked to develop some convertible bond tools for a London-based trading desk. Andreas Binder, CEO, of MathConsult managed to design a model solver based on Adaptive Integration in a one day workshop; a method based on Green's functions and an adaptive gridding and time-stepping.
When we decided to make UnRisk, we decided to integrate high-end numerical schemes into Mathematica and combine strengths of symbolic and numeric computation.