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 nonlinear problems, nonuniqueness 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.
