Christof Eck Harald Garcke - Mathematical Modeling

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Harald Garcke

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With (mathematical) modeling we denote the translation of a specific problem from the natural sciences (experimental physics, chemistry, biology, geosciences) or the social sciences, or from technology, into a well-defined mathematical problem. The mathematical problem may range in complexity from a single equation to a system of several equations, to an ordinary or partial differential equation or a system of such equations, to an optimization problem, where the state is described by one of the aforementioned equations. In more complicated cases we can also have a combination of the problems mentioned. A mathematical problem is well-posed , if it has a unique solution and if the solution of the problem depends continuously on its data, where continuity has to be measured in such a way that the results are meaningful for the application problem in mind. In general the phenomena to be described are very complex and it is not possible or sensible to take all its aspects into account in the process of modeling, because for example

Therefore nearly every model is based on simplifications and modeling assumptions . Typically the influence of unknown data are neglected, or only taken into account in an approximative fashion. Usually complex effects with only minor influences on the solution are neglected or strongly simplified. For example if the task consists of the computation of the ballistic trajectory of a soccer ball then it is sensible to use classical Newtonian mechanics without taking into account relativity theory. In principle using the latter one would be more precise, but the difference in results for a typical velocity of a soccer ball is negligible. In particular this holds true if one takes into account that there are errors in the data, for example slight variations in the size, the weight, and the kickoff velocity of the soccer ball. Typically available data are measured and therefore afflicted with measurement errors. Furthermore in this example certainly the gravitational force of the Earth has to be taken into account, but its dependence on the flight altitude can be neglected. In a similar way the influence of the rotation of the Earth can be neglected. On the other hand the influence of air resistance cannot be neglected. The negligible effects are exactly those which make the model equations more complex and require additional data, but do not improve the accuracy of the results significantly.

In deriving a model one should make oneself clear what is the question to be answered and which effects are of importance and have to be taken into account in any case and which effects are possibly negligible. The aim of the modeling therefore plays a decisive role. For example the model assumptions mentioned above are sensible for the flight trajectory of a soccer ball, but certainly not for the flight trajectory of a rocket in an orbit around the Earth. Another aspect shows the following example from weather forecasting: An exact model to compute the future weather for the next seven days from the data of today cannot serve for the purposes of weather forecast if the numerical solution of this model would need nine days of computing time of the strongest available supercomputer. Therefore often a balance between the accuracy required for the predictions of a model and the costs to achieve a solution is necessary. The costs can be measured for example by the time which is necessary to achieve a solution of the model and for numerical solutions also by the necessary computer capacities. Thus at least in industrial applications costs often mean financial costs. Because of these reasons there can be no clear separation between correct or false models, a given model can be sensible for certain applications and aims but not sensible for others.

An important question in the construction of models is: Does the mathematical structure of a model change by neglecting certain terms? For example in the initial value problem with the small parameter one could think about omitting the term . However, this would lead to an obviously unsolvable algebraic system of equations The term neglected is decisive for the mathematical structure of the problem independent of the smallness of parameter . Therefore sometimes terms which are identified as small, cannot be neglected. Hence, constructing a good mathematical model also means to take aspects of analysis (well-posedness) and numerics (costs) of the model into account.

The essential ingredients of a mathematical model are

The knowledge of the model assumptions is of importance to estimate the scope of applications and the accuracy of the predictions of a model. The aim of a good model is, starting from known but probably only estimated data and accepted laws of nature to give an answer as good as possible for a given question in an application field. A sensible model should only need data which are known or for which at least plausible approximations can be used. Therefore the task consists in extracting as much as possible information from known data.

To illustrate some important aspects of modeling in this section we consider a very simple example: A farmer has a herd of 200 cattle and he wants to increase this herd to 500 cattle, but only by natural growth, i.e., without buying additional animals. After a year the cattle herd has grown to 230 animals. He wants to estimate how long it lasts till he has reached his goal.

A sensible modeling assumption is the statement that the growth of the population depends on the size of the population, as a population of the double size should also have twice as much offspring. The data available are