|When studying any physical problem in Applied Mathematics, three essential stage are involved.
1. Modeling: An appropriate mathematical model, based on the
physics or the engineering of the situation, must be found. Usually these models are given a priori by the physicists or the engineers themselves. However, mathematicians can also play an important role in this process especially considering the increasing emphasis on non - linear models of physical problems.
2. Mathematical study of the model: A model usually involves a set of
ordinary’ or partial differential equations or an (energy) functional
to be minimized. One of the first tasks is to find a suitable functional space in which to study the problem. Then comes the study of existence and uniqueness or non -uniqueness of solutions. An important feature of linear theories is the existence of unique solutions depending continuously on the data (Hadamard’s definition of well - posed problems). But with non-linear problems, non-uniqueness is a prevalent phenomenon. For instance, bifurcation of solutions is of special interest.
3. Numerical analysis of the model: By this is meant the description of, and the mathematical analysis of, approximation schemes, which can be run on a computer in a ‘reasonable’ time to get ‘reasonably accurate’ answers.