Between quality and quantity

If mathematics is the language of science and engineering, differential equations form much of its grammar. A myriad of facts originating in the laboratory, in an astronomical observatory or on a field trip, flashes of enlightenment and sudden comprehension, the poetry of nature and the miracle of the human mind can all be phrased in the language of mathematical models coupling the behaviour of a physical phenomenon with its rate of change: differential equations. No wonder, therefore, that research into differential equations is so central to contemporary mathematics. Mathematical disciplines from functional analysis to algebraic geometry, from operator theory and harmonic analysis to differential geometry, algebraic topology, analytic function theory, spectral theory, nonlinear dynamical systems and beyond are, once you delve into their origins and ramifications, mostly concerned with adding insight into the great mystery of differential equations.

Modern mathematics is extraordinarily useful in deriving a wealth of qualitative information about differential equations, information that often has profound physical significance. Yet, except for particularly simple situations, it falls short of actually providing the solution in an explicit form. The task of fleshing out numbers on the mathematical bones falls to numerical analysis. And here looms danger … The standard rules of engagement of numerical analysis are simple: deploy computing power and algorithmic ingenuity to minimize error. Yet it is possible that, in our quest for the best quantity, we might sacrifice quality.