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The following diagram gives an idea of the process involved in
mathematical modeling.
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Real-world
Phenomenon
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Observation
Simplification |
Model
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 Trials |
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Analysis |
Real-world
Conclusions
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Interpretation |
Mathematical Conclusions
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We make specific observations about a phenomenon or behavior being
studied and identify factors that seems involved, making the
necessary simplifications. Next we conjecture tentative relations among
the factors we have selected creating a crude model of the phenomenon.
Having constructed a model, we then apply the appropriate
mathematical analysis leading to conclusions about the model. These
conclusions pertain only to the model and not the real-world system
under study.
Because the simplifications made in the construction of the model and
observations, the model will contain errors and limitations. We
must carefully account for these anomalies before drawing any
conclusions about the real-world behavior.
- Identify the Problem
This a difficult step because it is not clear what we need to
do. We need to sort through large amounts of data and identify
some particular aspect of the situation we want to study.
- Make
- Classify the Variables
List the things that influence the behavior you
identified in step 1 as variables. Identify dependent and
independent variables.
- Determine Interrelationships
Among the Variables Selected.
Before we can conjecture relationships among
the variables, we must make additional simplifications and
consider sub models if necessary.
- Solve the Model
The model now consist of mathematical equations or
inequalities that must be solved to find the information we
are seeking.
- Verify the Model
Before using the model, we need to test it. Does the model
answer the problem identified in step 1? Does the model make
common sense? Can we gather the data to operate the model? Be
careful about drawing conclusions and specially extrapolating
results!
- Implement the Model
Try to explain in terms that users of the model can understand
it. Help of a computer could be very useful.
- Maintain the Model
Remember that the model is derived form the specific problem
in step 1 and the assumption in step 2. Has the original
problems changed in any way? Do we need to consider some of
the neglected factors?
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