6-мавзу. Математик моделлаштириш элемент­лари

a) From the point of view of the system's evolution, the models are:

1. 
   in stationary regime (steady state in hydraulics);
   
2. 
   in dynamic regime (non-steady state in hydraulics).
   

Generally, the systems have an evolution in time and they have a dynamic character. However, there are situations in which one may use stationary models (for which the input variables, the system's states and the output variables are constant, being not variable in time).
Let's consider for example an aquifer for which the input and output are supposed to be constant (average multi-annual percolation, respectively multi-annual values of abstracted water through waterworks). The stationary model allows to determine the hydraulic potential of the aquifer, supposed invariant in time. Similar considerations may be made referring to the permanent movement in natural waterbeds, channels or pipes.
The stationary model allows, thus the determination of the balance conditions of the system, which is supposed to be stable, for constant inputs.
In case the variables depend on the time, the model is dynamic; variable input determine variable output, as well as the evolution of the system's states. The variables describe trajectories, the links between variables introducing links between the trajectories.
In general, the study of the system's dynamic is realized in the past-future direction, although in principle the examination from present (or future) towards past is also possible.


b) Considering the mathematic nature of the relations used in the model's formulation, the following models stand out:

1. 
   linear models;
   
2. 
   non-linear models.
   

The model is linear when the variables are at the first power and non-linear in a contrary case (at least one relation contains a variable at a degree different from one).
In most of the cases (and especially for natural systems), the relation between cause and effect is non-linear. However, for practical studies linear relations between variables are largely accepted, due to the advantages involved by using the principle of the effects superposition. This principle is based on the following two hypotheses:

• 
  The system's output is proportional with the input's intensity (the proportionality property).
  
• 
  In the case of a complex input, which may be decomposed in more partial inputs, the system's output is obtained through the superposition of the partial outputs produced by these partial inputs (the additivity property).