MODELING AND SIMULATION
What is a mathematical model?
Mathematical models are a way of analyzing systems by using equations (mathemati-
cal language) to determine how the system changes from one state to the next (usual-
ly using differential equations) and/or how one variable depends on the value or state
of other variables. Simply put (although working through the equations is not always
simple), the process of mathematical modeling includes the input of data and infor-
mation, a way of processing the information, and an output of results.A mathematical model can describe the behavior of many systems, including sys-
tems in the fields of biology, economics, social science, electrical and mechanical
engineering, and thermodynamics. For example, modeling is usually used in the sci-
ences to better understand physical phenomena; each phenomenon is translated into
a set of equations that describe it. But don’t think that all the results of models are
indicative of the real world. Because it is virtually impossible to describe a phenome-
non totally, models are considered to be merely a human construct to help us better
understand our surrounding real-world systems.What are some basic stepsto buildinga mathematical model?
Just like building a physical structure, there are basic steps to building a mathemati-
cal model. The first steps are to simplify the assumptions, or to clearly state those
assumptions on which the model will be based, including an understandable account
of the relationships among the quantities to be analyzed. Second is to describe all the
variables and parameters to be used in the model, and identify the initial conditions of
the model. Finally, use step one’s assumptions and step two’s parameters and variables
266 to derive mathematical equations.
What are some types of mathematical models?
M
athematical models are commonly broken down into numerical or analyti-
cal models. Numerical models are those that use some type of numerical
timing procedures to figure out a model’s behavior over time. The solution is
usually represented by a table or graph. An analytical model usually has a closed
solution. In other words, the solution to the equations used to describe changes
in the system can be expressed as a mathematical analytic function. Mathematical
models can also be divided in other ways. For example, some models are consid-
ered deterministic (or a model that always performs the same way for a given set
of initial conditions) or stochastic (or a model in which randomness is present).