A mathematical model uses mathematical notation for describing a system, for example from physics, biology or the social sciences, and thus allows the systematic exploration of the topic with mathematical methods. The creation process is called modeling. By comparing the calculation results with a model of the accuracy of observation can be verified.
Examples of applications of mathematical models are about predictions of climate change, weather, or the proof of stability ( static ) of a building.
- 5.1 Formulation of the model
- 5.2 Test of the model
- 5.3 Validation of the model
- 6.1 Electrical
- 6.2 Physics
- 6.3 Astronomy
- 6.4 Chemistry
- 6.5 Mathematics
- 6.6 Game theory ( economics )
- 6.7 Business Administration
- 6.8 sociology
Emergence and dissemination of the concept model
That model concepts play an increasingly important role in scientific theory formation has been clearly recognized in the discussion of atomic models in the early 20th century. Due to the epistemological role models of physics, the concept model, like other original physical concepts also spread to other disciplines.
Model-based methods are not limited to the natural sciences. For example, based the known two-dimensional plots of functional relationships in economics in radically simplifying modeling.
Concepts in modeling
Main article for systems within the systems theory: Systems theory
Mathematical models model systems.
In simple terms, a system described as a set of objects that are connected by relations. A system can be (such as a motor or a bridge), but also a virtual system (such as the logic of a computer game ) is a natural system (such as a lake, a forest ), a technical system.
A system is enclosed by its environment. This environment acts from outside the system. Such agents are referred to as relations. A system responds to impacts due to changes in system variables.
Basically, a system also has effects outwards, ie to the environment. In the context of modeling systems, this outward effect is however usually negligible.
A system is sealed from the environment by clearly defined system boundaries. This means that in modeling the defined relations are exclusively active. As an example, the study of the phosphorus discharges was called into a lake. In the context of a model to the only source opens into the lake flow are considered, the limit of the system in this example is then the relation " river ". More emergent in nature sources (groundwater, waterways, fish, and so on ) are not considered in the model.
The definition of a concrete system as a subject in the modeling of mathematical models made by the analyst according to the study objective.
Schematically, a system can be a so-called box model pose.
The box is the modeled system. The input relation are symbolic of the impact of the environment on the modeled system and the outgoing arrow symobilisiert the changes of the system. Between the system variables themselves, any other relations can exist.
In practice, box models are used as food aid. The graphical representation of a system simplifies the detection of system variables. A modeled system may be composed of any number of other subsystems that again constitute a separate box model, respectively.
The box model is used in particular in the engineering sciences in the creation of computer models. The models each represent a total box model (more precisely the relations within the system ) dar. Each graphic element is again a separate box model. Different graphical symbols for box models were used to simplify this, such as a coiled symbol to represent the system variable of a coil.
As part of the modeling systems are feasible, have the effects to the outside, but no input relations. About a system which produces clock pulses. It also systems are conceivable which via input ratios but do not have effects. For example, the monitoring of values.
According to the degree of certainty of a box model is in Black box models can be distinguished box and white box models. Black box models describe the behavior of a system in the form of an equation, without taking into account the complexity of the system. White box models try on the other hand a system as accurately as possible to model.
The choice of these models depends on the object of investigation. If only serve as a mathematical model calculation tool, a black box is sufficient. If the internal behavior of a system, such as a simulation to investigate a white box must be created by force.
Main article for physical dimensions: dimension (size system ). Main article for mathematical dimensions: dimension (mathematics)
The dimension of the system is the number of state variables, with the mathematical model will be described.
A model equation is the formal mathematical model of a system in the form of a function.
Model equations basically have the form
Basically, each of the sets may be empty. Often the amount is only one element. It is therefore customary to specify only the required quantities in a concrete model equation and to determine the elements of the required subsets by index. Depending on the field of science for a model equation is created, get the elements of a model equation other variable names.
In the various fields of different variable names are used to match the technical jargon and common variable name of the area.
Types of mathematical models
Classification of mathematical models
Modeling of an example system
Formulation of the model
Magnetism can have several causes; in a single magnet various mechanisms have impact, produce, strengthen or weaken the magnetism; the magnet can consist of a complex design, contaminated materials; and so on. In this mess you're trying to bring light, by studying model systems. A physical model for a ferromagnet can go something like this: an infinitely extended ( one sees of surface effects from ), periodic ( one sees from lattice defects and impurities from ) arrangement of atomic dipoles ( one focuses on the magnetism of bound electrons and describes this in the simplest mathematical approximation).
Examination of the model
In order to investigate the just- introduced physical model of a ferromagnet, different methods are possible:
- You could build a three-dimensional, physical model, such as a wooden lattice ( representing the atomic lattice ), in the floating bar magnet ( representing the atomic dipoles ) are suspended. One could then investigate experimentally how they affect the bar magnets in their orientation to each other.
- Since the laws of nature, which the atomic dipoles are subjected, are well known, but you can also describe the model magnet through a system of closed equations: in this way has been obtained from the physical model of a mathematical model. This mathematical model can be solved exactly or asymptotically in favorable cases with analytical methods.
- In many cases, uses a computer to analyze a mathematical model numerical.
- In the above example, one can vary the physical location of the dipoles or their interaction desired. Thus, the model loses the claim to describe a reality; one is interested now as to which mathematical consequences of a change in the physical assumptions.
Validation of the model
You select parameters that one hand knows from experimental studies in real ferromagnets and the one other hand, can also be determined for the model; in the specific example, for example, the magnetic susceptibility as a function of temperature. If a model and model coincide in this parameter, then back conclude that the model reproduces the relevant aspects of reality correctly.
Examples of mathematical models
Game theory ( economics )
Limits of mathematical models
"After having been Involved in Numerous modeling and simulation efforts, Which produced far less than the Desired results, the nagging question Becomes; Why? "
"After I have participated in many modeling and simulation efforts that have brought far less than the hoped-for results, the nagging question arises; Why? "
Mathematical models are a simplified representation of reality, not reality itself serve the study of specific aspects of a complex system and make sure simplifications in purchasing. In many areas, a complete modeling of all variables would lead to an unmanageable complexity. Models, especially those that describe human behavior, provide only an approximation to reality dar. It is not always possible with models to make the future predictable.