Modeling via Differential Equations

One of the most difficult problems that a scientist deals with in his everyday research is: "How do I translate a physical phenomenon into a set of equations which describes it?''

It is usually impossible to describe a phenomenon totally, so one usually strives for a set of equations which describes the physical system approximately and adequately.

In general, once we have built a set of equations, we compare the data generated by the equations with real data collected from the system (by measurement). If the two sets of data "agree'' (or are close), then we gain confidence that the set of equations will lead to a good description of the real-world system. For example, we may use the equations to make predictions about the long-term behavior of the system. It is also important to keep in mind that the set of equations stays only "valid" as long as the two sets of data are close. If a prediction from the equations leads to some conclusions which are by no means close to the real-world future behavior, then we should modify and "correct" the underlying equations. As you can see, the problem of generating "good" equations is not an easy exercise.

Note that the set of equations is called a Model for the system.

How do we build a Model?

The basic steps in building a model are:

Step 1: Clearly state the assumptions on which the model will be based. These assumptions should describe the relationships among the quantities to be studied.

Step 2: Completely describe the parameters and variables to be used in the model.

Step 3: Use the assumptions (from Step 1) to derive mathematical equations relating the parameters and variables (from Step 2).

The best example of mathematical modeling is the one related to population growth problems. Keep in mind that this problem has many ramifications ranging from population explosion to extinction phenomena.

[Differential Equations] [Calculus]
[Algebra] [Trigonometry ]
[Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi
Last Update 6/22/98

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