Applied mathematics is currently understood as a set of tools and methods designed to solve a wide variety of tasks facing humanity. This can be both the tasks of scientific cognition, and the tasks of developing new technical devices, and the tasks of creating new technological processes, and the tasks of management and optimization in a wide variety of human activities, including the management of society.
In recent history the possibilities of applied mathematics were convincingly shown, for instance, in solving the most important problems connected with development of nuclear energy and creation of rocket-space technologies.
The activity of the applied mathematician usually takes place in close cooperation with a specialist in a particular field (physicist, chemist, biologist, engineer, manager, etc., or, as it is usually called, the customer). Briefly, this activity is characterized by the motto: model – algorithm – program – model (A.A. Samarsky).
On one hand, development of a mathematical model is based on general scientific ideas about an object or process under study and, on the other hand, an efficient model must be based on modern means of fundamental mathematics, so, at this stage, the role of a mathematician is of great importance.
As a rule, a mathematical model is a complex equations or some other mathematical objects. Direct solution by analytical, exact, methods of the corresponding problems usually turns out to be impossible. Previously it is necessary to perform their so-called discretization, allowing approximately, but with sufficient accuracy to reduce the problem to simpler problems, for example, systems of linear algebraic equations. This is the initial stage of algorithmization.
It is clear, that realization of algorithm can be executed only by means of computer, after development of the corresponding program, and usually a complex of programs which includes the so-called intellectual kernel realizing the algorithms of the problem solution, and also the interactive means providing input of information, for example, from corresponding measuring devices, from databases, etc., and also output of information in the form convenient for the customer (tables, databases, graphics, video, sound, etc.).
As a rule, in the course of operation of a program complex, there is a need of the subsequent specification of model, algorithms, programs, and such cyclic joint work of the mathematician-applicator and the customer usually proceeds all the time while the solved problem remains actual.