Mathematical model

A mathematical model is a means of mathematical notation generated model to describe a section of the observable world. This model can be used in any limited areas of observable reality, such as B. the natural sciences, economics or social sciences, medicine or engineering. Mathematical models allow a logical, structural penetration depending on the type with regard to valid laws, allowed and not allowed states, as well as its dynamics with the aim of transferring this knowledge to the modeled system.

The process of creating a model is called modeling . The creation of a mathematical model for a section of reality is no longer the task of mathematics, but of the respective scientific field. The extent to which a mathematical model correctly describes processes in reality must be checked and validated through measurements.

A mathematical model thus establishes a reference to reality that does not generally have to be present for mathematical sub-areas.

Appearance and spread of the term model

That model ideas play an increasingly important role in the formation of scientific theories was clearly recognized in the discussion of atomic models at the beginning of the 20th century. Due to the epistemological role model function of physics, the term model, like other originally physical terms, has spread to other disciplines.

Model-based methods are not limited to the natural sciences . For example, the well-known two-dimensional plots of functional relationships in economics are based on radically simplified modeling.

Terms in modeling

system

Main article for systems within systems theory: systems theory

Mathematical models model systems.

In simple terms, a system can be described as a set of objects that are connected by relations . A system can be a natural system (e.g. a lake, a forest), a technical system (e.g. a motor or a bridge), but also a virtual system (e.g. the logic of a computer game).

A system is enveloped by its environment. This environment affects the system from the outside. Such effects are called relations . A system reacts to the effects of changes in system variables. ${\ displaystyle {\ mathcal {R}}}$

In principle, a system also has external effects, i.e. on the environment. In the context of the modeling of systems, however, this outward-looking effect is usually neglected.

A system is closed off from the environment by clearly defined system boundaries. This means that only the defined relationships are effective for modeling. An example is the investigation of the phosphorus input into a lake. In the context of a model, a river flowing into the lake is to be considered as the only source; the limit of the system in this example is the relation “river”. Other natural sources (groundwater, shipping, fish, and so on) are not taken into account in the model.

The definition of a specific system as the object of investigation when modeling mathematical models is carried out by the analyst in accordance with the aim of the investigation.

Box model

A system can be represented schematically using a so-called box model:

${\ displaystyle {\ mathcal {R}} \ \ rightarrow \ {\ begin {array} {| c |} \ hline \ quad \\\ quad {\ mathcal {V}} _ {i} \ leftrightarrow {\ mathcal { V}} _ {n} \ quad \\\ quad \\\ hline \ end {array}} \ \ rightarrow}$

The box represents the modeled system. The input relation symbolizes the effects of the environment on the modeled system and the outgoing arrow its reactions. Any further relationships can exist between the system variables themselves. ${\ displaystyle {\ mathcal {R}}}$ ${\ displaystyle {\ mathcal {V}}}$

In practice, box models are used as a thinking aid. The graphical representation of a system simplifies the identification of system variables. A modeled system can consist of any number of additional subsystems, each of which represents its own box model.

The box model is used particularly in engineering when creating computer models. The models represent a total of a box model (more precisely the relationships within the system). Each graphic element is in turn its own box model. To make things easier, different graphic symbols were used for box models, such as a coiled symbol to represent the system variables of a coil.

In the context of modeling, systems are conceivable that have external effects, but no input relations. For example a system that produces time cycles. Systems are also conceivable that have input relations but do not have effects. For example, to monitor values.

According to the degree of specificity of a box model, box models can be divided into black 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, on the other hand, try to model a system as precisely as possible.

The choice of one of these models depends on the subject of the investigation. If a mathematical model is only to serve as a calculation aid, a black box is sufficient. If the internal behavior of a system is to be examined , for example in a simulation , a white box must be created.

dimension

Main article for physical dimensions: Dimension (size system) . Main article for math dimensions: Dimension (math)

The dimension of a system is the number of state variables with which the mathematical model is described.

Model equation

A model equation is the formal mathematical model of a system in the form of a function .

Model equations basically have the form ${\ displaystyle \ {{\ mathcal {V}} \} = f (\ {{\ mathcal {R}} \}, \ {{\ mathcal {V}} \}, \ {p \})}$

${\ displaystyle \ {{\ mathcal {V}} \}}$: is a set of model variables.
${\ displaystyle \ {{\ mathcal {R}} \}}$: is a set of relations that affect the system.
${\ displaystyle \ {p \}}$: is a set of model constants.

In principle, each of the quantities can be empty. Often the set consists of just one element. It is therefore customary to specify only required quantities in a specific model equation and to determine the elements of the required subsets using an index. Depending on the scientific area for which a model equation is created, the elements of a model equation are given different variable names.

Different variable names are used in the different subject areas , matching the technical terminology and common variable names of the area.

Types of mathematical models

Static models

Describe the state of a system before and after changes to external relations, but not during a change. A simple static model would be the calculation of the mixing temperature of two different warm liquids. Using a static model, the temperature before mixing can be calculated and the temperature after mixing can be calculated. The system equation of a static model has the general form

{\ displaystyle {\ mathcal {V}} = {\ mathit {f}} ({\ mathcal {R}})}

where any complex function can be and it is entirely possible to pass further parameters to this function as constants.

{\ displaystyle {\ mathit {f}}}

Dynamic models

Describe the reaction of a system to changes in external relations. Such models could be used to describe the temperature change of the mixture during the mixing.

Continuous models over time

Describe the reaction of a system to changes in external relations over a continuous period of time. The modeling is done with differential equations . In the mixing example, the model would be a function that can be used to calculate the change tendency at any point in time. The temperature can be calculated at any point in time by integrating the equation. The system equation of a temporally continuous model has the general form: Temporal change of or

{\ displaystyle {\ mathcal {V}} = {\ mathit {f}} ({\ mathcal {R}}, {\ mathcal {V}})}

{\ displaystyle {\ frac {\ rm {d {\ mathcal {V}}}} {\ rm {d {\ mathit {t}}}}} = {\ mathcal {V}} (t) '= {\ mathit {f}} ({\ mathcal {R}}, {\ mathcal {V}})}

Temporally discrete models

Not all processes can be described continuously. Measurements are often only made at certain intervals. The system status between these intervals is not known, i.e. it is discrete. With the help of time series analyzes , difference equations can be created for modeling such systems. The system equation of such a model has the general form

{\ displaystyle {\ mathcal {V}} ^ {(k + 1)} = {\ mathit {f}} ({\ mathcal {R}} ^ {(k)}, {\ mathcal {V}} ^ { (k)})}

Spatially continuous models

To model systems in which the spatial dimension is relevant in addition to the temporal dimension, spatially continuous models are created with the help of partial differential equations . In the mixing example, such a model could be used, for example, to determine which temperature is reached at a certain point in time at a certain point in the mixing vessel.

Stochastic models

Not all systems behave deterministically, i.e. predictably. A typical example is the decay process of radioactive isotopes. Over a period of time, a certain amount of isotopes can be expected to decay, but it is not possible to predict when exactly this will happen. To model such systems, stochastic models are created with the help of probability calculations.

Classification of mathematical models

After the change behavior

into static or dynamic models

According to continuity

into discrete or continuous models (see also grid model )

According to predictability

into stochastic and deterministic models

According to the number of system variables

in 1- to n -dimensional models

According to the type of equations and systems of equations

into linear, quadratic and exponential models

According to the branch of science

for example in physical, chemical, ... models

Modeling an example system

Formulation of the model

Magnetism can have different causes; Various mechanisms can act in a single magnet, producing, strengthening or weakening magnetism; the magnet can consist of complex, contaminated materials; and so on. One tries to shed light on this mess by examining model systems. A physical model for a ferromagnet can be something like this: an infinitely extended (one disregards surface effects), periodic (one disregards lattice defects and impurities) arrangement of atomic dipoles (one concentrates on the magnetism of bound electrons and describes it in the simplest mathematical approximation).

Examination of the model

Various methods are conceivable to investigate the physical model of a ferromagnet that has just been introduced:

One could build a three-dimensional, physical model, such as a wooden lattice (which represents the atomic lattice), in which freely movable bar magnets (which represent the atomic dipoles) are suspended. Then one could investigate experimentally how the bar magnets influence each other in their alignment.
Since the laws of nature to which the atomic dipoles are subject are well known, the model magnet can also be described by a system of closed equations: in this way a mathematical model has been obtained from the physical model.
- In favorable cases, this mathematical model can be solved exactly or asymptotically using analytical methods.
- In many cases, a computer is used to numerically evaluate a mathematical model.
A so-called computer model is nothing more than a mathematical model that you evaluate with a computer. This process is also called computer simulation .
The investigation of models can, like any scientific activity, become independent:
- In the physical example mentioned, the arrangement of the dipoles or their interaction can be varied as desired. The model thus loses its claim to describe a reality; one is now interested in the mathematical consequences of a change in the physical assumptions.

Validation of the model

One chooses parameters that one knows from experimental investigations on real ferromagnets and that one can also determine for the model; in a specific example, for example, the magnetic susceptibility as a function of temperature. If the prototype and the model agree in this parameter, then one can conclude that the model correctly reflects relevant aspects of reality.

Examples of mathematical models

Probably the best-known and oldest application examples for mathematical models are the natural numbers, which describe the laws of “counting” concrete objects, the extended number models, which describe classic “computing”, and the geometry that made land measurement possible.

Electrical engineering

Electrical resistance of a conductor : Based on Ohm's law, this model is used to calculate the value of an ideal ohmic resistance. This is a one-dimensional, static model.

physics

Rocket equation : This model describes the principles of rocket propulsion. It is a one-dimensional, time-continuous model. The variable is the outflow velocity, while the rocket and propellant mass are only parameters.

astronomy

Law of Gravitation : Newton's Law of Gravitation is a three-dimensional, space-continuous model.

chemistry

Gibbs-Helmholtz equation : This model describes the heat balance of chemical reactions. It is a time-discrete model.

mathematics

Galton board : The Galton board is an experimental set-up to illustrate probability distributions. The model is an example of a one-dimensional, stochastic model.

Game Theory (Economics)

Prisoner's Dilemma : This model describes how two subjects make a rationally advantageous decision that is generally disadvantageous for both sides. It is a two-dimensional, time-discrete model.

Business administration

Profit maximization : Depending on the cost structure and the sales function, the point with the maximum profit can be calculated with this model. It is a two-dimensional, time-continuous model.

sociology

La Ola wave: The mathematical model of the La Ola wave is described in nature magazine.

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 participating in many modeling and simulation efforts that have produced far less than what I had hoped for, the nagging question arises; Why?"

- Gene Bellinger : Mental Model Musings

Mathematical models are a simplified representation of reality, not reality itself. They are used to investigate partial aspects of a complex system and accept simplifications in return. In many areas, a complete modeling of all variables would lead to a complexity that is no longer manageable. Models, especially those that describe human behavior, only represent an approximation of reality. It is not always possible to use models to make the future predictable.

literature

Dieter M. Imboden, Sabine Koch: System analysis: Introduction to the mathematical modeling of natural systems. 3. Edition. Berlin 2008, ISBN 978-3-540-43935-6 .
Claus Peter Ortlieb , Caroline von Dresky, Ingenuin Gasser, Silke Günzel: Mathematical Modeling - An Introduction to Twelve Case Studies. 2nd Edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-00534-4 .
Frank Haußer, Yury Luchko: Mathematical Modeling with MATLAB® - A Practice-Oriented Introduction. Spectrum Akademischer Verlag, Heidelberg 2011, ISBN 978-3-8274-2398-6 .
Christof Eck , Harald Garcke , Peter Knabner : Mathematical Modeling. 3. Edition. Springer, 2017, ISBN 978-3-662-54334-4 .

Web links

Individual evidence

↑ Dieter M. Imboden, Sabine Koch: System Analysis: Introduction to the Mathematical Modeling of Natural Systems. 3. Edition. Berlin 2008, ISBN 978-3-540-43935-6 .
↑ Examples of computer models created using the box model at maplesoft.com. Retrieved December 27, 2009 .
^ I. Farkas, D. Helbing & T. Vicsek: Social behavior: Mexican waves in an excitable medium. In: nature 419. Nature Publishing Group, a division of Macmillan Publishers Limited, September 12, 2002, pp. 131-132 , accessed December 27, 2009 : “The Mexican wave, or La Ola, which rose to fame during the 1986 World Cup in Mexico, surges through the rows of spectators in a stadium as those in one section leap to their feet with their arms up, and then sit down again as the next section rises to repeat the motion. To interpret and quantify this collective human behavior, we have used a variant of models that were originally developed to describe excitable media such as cardiac tissue. Modeling the reaction of the crowd to attempts to trigger the wave reveals how this phenomenon is stimulated, and may prove useful in controlling events that involve groups of excited people. "
↑ Andrea Naica-Loebell: Mathematical model of the La Ola wave. In: Telepolis, heise online. September 15, 2002, accessed on December 27, 2009 : “Crowds react like particles from chemistry. In the current issue of the science magazine Nature, a computer simulation is presented that explains the mechanisms of the La Ola wave in football stadiums. "
^ Gene Bellinger: Simulation Is Not The Answer. In: Mental Model Musings. 2004, accessed December 27, 2009 .

[1] Dieter M. Imboden, Sabine Koch: System Analysis: Introduction to the Mathematical Modeling of Natural Systems. 3. Edition. Berlin 2008, ISBN 978-3-540-43935-6 .

[2] Examples of computer models created using the box model at maplesoft.com. Retrieved December 27, 2009 .

[3] I. Farkas, D. Helbing & T. Vicsek: Social behavior: Mexican waves in an excitable medium. In: nature 419. Nature Publishing Group, a division of Macmillan Publishers Limited, September 12, 2002, pp. 131-132 , accessed December 27, 2009 : “The Mexican wave, or La Ola, which rose to fame during the 1986 World Cup in Mexico, surges through the rows of spectators in a stadium as those in one section leap to their feet with their arms up, and then sit down again as the next section rises to repeat the motion. To interpret and quantify this collective human behavior, we have used a variant of models that were originally developed to describe excitable media such as cardiac tissue. Modeling the reaction of the crowd to attempts to trigger the wave reveals how this phenomenon is stimulated, and may prove useful in controlling events that involve groups of excited people. "

[4] Andrea Naica-Loebell: Mathematical model of the La Ola wave. In: Telepolis, heise online. September 15, 2002, accessed on December 27, 2009 : “Crowds react like particles from chemistry. In the current issue of the science magazine Nature, a computer simulation is presented that explains the mechanisms of the La Ola wave in football stadiums. "

[5] Gene Bellinger: Simulation Is Not The Answer. In: Mental Model Musings. 2004, accessed December 27, 2009 .