Mathematical modelling of Systems
The use of the Mathematical “word Applied” is, in many, to refer to methods and tools that apply in the resolution of problems in the basic sciences, applied and in the social sciences. It is very known how the mathematical methods have been efficient for the study of numerous problems of the physics, biology, chemical, medicine, social sciences, engineering, computer ecology, economy etc. The transversalidad, in relation to his applications, can consider one of his main virtues. In reality any area of the mathematical can be used for the analysis and resolution of real problems, but in the mathematics applied pursues his application and transfer to other areas. The mathematics applied are very used in the technologies for the study, modelling, simulation and optimisation of phenomena as well as in the design of experiments. They are diverse the areas inside the mathematics that have numerous applications to other sciences: linear algebra, functional analysis, complex variable, analysis of Fourier, numerical analysis, calculation, computation, differential equations, statistical, operators, discreet mathematics, optimisation, dynamic systems, probability, theory of control, etc. In the case of the mathematics applied is obvious that it is necessary to include how important part and head office in his development the area that covers the numerical analysis and the scientific computation, but without establishing limits or borders to the equal that occurs in other disciplines.
It has observed through the last times that the application and methodology in the mathematics has experienced a big change and the phenomena considered in the industry are analysed with the scientific use of the calculation and the numerical simulation, indispensable tools for the analysis of the processes. It is therefore very important the mathematical modelling for the resolution of scientific problems that comport some needs of sophisticated calculation.
Of generic form, in sciences applied, a mathematical model is a type of scientific model that employs a mathematical methodology to express relations, proposiciones substantive of facts, variables, parameters, entities and relations between variables and/or entities or operations, and that studies the behaviours of the systems in front of difficult situations to observe in the reality. Besides the term mathematical modelling is used also in graphic design when it speaks of geometrical models of the objects in two (2D) or three dimensions (3D).
The meaning of mathematical model in philosophy of the mathematics and foundations of the mathematical is, however, something different. In concrete in these areas work with "formal models". A formal model for one some mathematical theory is a group on which have defined a group of relations unarias, binary and trinarias, that satisfies the proposiciones derivative of the group of axioms of the theory. The branch of the mathematics that commissions to study systematically the properties of the models is the theory of models.
To the equal that all type of scientific investigation, the modelling of a system, as they are the complex systems, begins with the definition of the problem or phenomenon to analyse and for which has to fix some aims. Nevertheless this starting point does not have why be the ideal and begun the process of modelling, and during the process, is very likely that owe to modify the limits established returning again to the starting point. This is had to, in some measure, to the need of simplification without that thus they exclude the elements of importance and interest in the practical investigation. Often the problem estriba in finding the balance between the simplification and the own complexity of what studies , since it is necessary to find a solution that was appropriate and reflect the process object of study. On the other hand it is important the extension so much in the time as in the space, or said of another form in general wishes that the investigation can apply to the possible maximum of situations and in the greater period of time, what can be a problem for the delimitations that have marked .
In the actuality is difficult to present an only classification of the models since there is a big number of classifications that refer to different appearances. A classification is the one who divides to the models in determinísticos (the same entrances produce always the same exits, the chance does not play any paper) and stochastic/probabilistic (the chance plays a paper, a same entrance can produce diverse states and exits) depending if they use probabilities and in consequence if the results are determinate or approximated with some uncertainty. On the other hand if it has in consideration that include or no the time in the model, the classification is in dynamic or static models. But when taking into account the variable time distinguishes between a discreet or continuous model depending on that the variable time consider in discreet intervals: years, months or of continuous form. The models that implement in a computer will be discreet and if the model is of continuous origin it is necessary to realise a process of discretisation to approximate the continuous model by one discreet in a finite number of values (for example inside the numerical calculation). Obviously it will be able to speak of dynamic models that they are stochastic, as for example models of prediction of the time.
It could also speak of “mathematical model” like that model that uses language and mathematical methods.
In the modelling of complex systems in which the number of variables and relations can be elevated is of vital importance the obtaining of methodologies that help us to modelar relations, between which find those relations of which only know experimental data.
Universidad de Alicante
Carretera de San Vicente del Raspeig s/n
03690 San Vicente del Raspeig
Tel: (+34) 96 590 3691Fax: (+34) 96 590 3464