A Study of the Mathematical Models related to Diffusion Dispersion

First it should be said that the term “mathematical modelling” took a tremendous leap in popularity some 35 years ago. There are now journals of mathematical modelling in various forms and an International Society to canonize an activity that has been going on for many years. The essential quality in a model is that it should be capable of having a life of its own. It may not, in practice, need to be sundered from its physical matrix. How powerful the life that is instinct in a true mathematical model can be seen from the Fourier’s theory of heat conduction, where the mathematical equations are fecund of all manner of purely mathematical developments. At the other end of the scale a model can cease to be a model by becoming too large and too detailed a simulation of a situation whose natural tine of development is to the particular rather than to the general. It ceases to have a life of its own by becoming dependent for its vitality on its physical realization. Mathematical modelling, thus, is the art and craft of devising a mathematical structure that will most elegantly penetrate to the essence of a given situation.