Modelling

“How do we translate a physical phenomenon into a set of equations which describes it?” – this is certainly one of the most difficult problems that scientists deal with in their everyday research. This problem is a difficult one since it is usually impossible to describe a phenomenon totally, so one often tries to reformulate a real-world problem as a mathematical one making certain simplifying assumptions. As a result, one usually describes the system approximately and adequately. The problem of generating “good” equations is not an easy task. The set of equations one deals with is called a model for the system. In general, once we have built a set of equations, we compare the data generated by the equations with real data collected from the system (by measurement). When the two sets of data “agree” (or are “sufficiently” close), we gain confidence that the set of equations will lead to a good description of the real-world system. If a prediction from the equations leads to some conclusions which are by no means close to real-world future behavior, we should modify and “correct” the underlying equations. When creating a model, it is necessary to formulate the problem under consideration into questions that can be answered mathematically.

The following are basic steps in building a model.

(1) Clearly state the assumptions on which the model will be based. These assumptions should describe the relationships between the quantities to be studied.

(2) Completely describe the parameters and variables to be used in the model.

(3) Use the assumptions (from step (1)) to derive mathematical equations relating the parameters and variables (from step (2)).

A large number of laws of physics, chemistry, economics, medicine, etc. can be formulated as differential equations. They serve as models that describe the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. Whenever a mathematical model involves the rate of change of one variable with respect to another, a differential equation is apt to appear. As we will see in the following sections, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, the mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. In the following sections we provide examples of mathematical models of several relatively simple phenomena which are described by ordinary differential equations of the form (1). We mainly concentrate on modeling issues. The modern theory of ordinary differential equations together with known techniques, methods and applications can be found in references [1] and [2].

Computer simulation

A computer simulation (also referred to as a computer model or a computational model) is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system. Computer simulations have become a useful part of the mathematical modelling of many natural systems in physics (computational physics), chemistry and biology; human systems in economics, psychology, and social science, and in the process of engineering new technology, so as to gain insight into the operation of those systems or to observe their behaviour.

While computer simulations might use some algorithms from purely mathematical models, computers can combine simulations with the reality of actual events, such as generating input responses to simulate test subjects who are no longer present. Although the missing test subjects (i.e. the users of equipment or systems) are being mathematical model an abstract model that uses mathematical language to describe a system modelled/simulated, the whole process can be conducted with the actual equipment or system they use, revealing performance limits or defects in long-term use by the simulated users. -Note that the term computer simulation is broader than computer modelling, which implies that all aspects are being modelled in the computer representation. However, computer simulation also includes generating inputs from simulated users to run actual computer software or equipment, with only part of the system being modelled: an example would be flight simulators which can run machines as well as actual flight software.