6-мавзу. Математик моделлаштириш элемент­лари

Physical models (also called imitative or iconic models) resemble physically to the object they represent, but they are presented on a reduced scale. In this category we find, for example, hydraulic or aerodynamic models, the models used in architecture etc. Special categories of the physical models are the analogical models, which use a domain's properties, to model another domain's properties; for example, the infiltration through earth dams was studied using electric fields properties.
The abstract models (mathematical or symbolic models) represent unitary systems of variables and mathematical relations destined to the analysis of a part of the reality, serving to the discovery of new ways of organization and behaviour, which cannot be perceived by other means.
Real phenomena depend of a great number of variables, between which there are certain relations; still, not all of these variables have the same importance. The mathematical model represents a simplified mathematical description of a phenomenon or process, detaching from the large number of variables, those that intervene essentially; thus a satisfying approximation of the reality on the basis of a reduced number of variables and relations among them is realized.
The model's elements are mathematical relations, which may be algebraic, or differential equations or systems of equations (reflecting the links between the system's components or their behaviour), as well as inequalities, specifying the limits between which the variables' values may be situated. The coefficients of these relations, known as models' parameters, depend on the investigated system structure and generally are estimated during a calibration process.
The model's variables are of two types:

1. 
   decision variables, representing unknown values, which are to be determined during the mathematical modelling;
   
2. 
   state variables, which characterize the system's evolution and which depend on the input and the decision variables.
   

In current practice, one must find a balance between the wish to create a complex instrument for investigation, which will describe as exactly as possible the evolution of the studied system and the imperative that the model should be efficient, meaning to find the solutions in a reasonable time and providing good results.
Practical experience has actually demonstrated that there cannot be only one mathematical model to contain all the important aspects of the modelled system; the process which is used consists of creating models specialized in solving some specific problems.
The complexity of the used model itself depends on the accuracy and volume of the input data in the model. Like this, the use of some approximate data for an exact model usually leads to important errors, the application of a simplified model being preferable, as it will require less primary information. The precision of the model and the precision of the input data are thus in a relation of tight interdependency. For example, if for the flood waves routing the geometry and the hydraulic characteristics of the riverbed are known, the Saint-Vénant model may be used; otherwise, hydrological models like Muskingum are indicated.
The elaboration of a model in general and of a mathematical model specifically, may be realized either by starting from scientific knowledge already existent, either from measurements over some phenomena and processes towards a theoretical approach. In other words, the model represents an instrument of knowledge, a method of verifying the theory, but also an important stage in the formulation of a new theory.

2. Моделлар таснифи.

7.3. Classification of the mathematical models
In function of the criteria considered the following classification of mathematical models results: