2.3.5 MATHEMATICAL MODELING

 

Mathematical modeling is the process of creating a mathematical representation of some phenomenon in order to gain a better understanding of that phenomenon. It is a process that attempts to match observation with symbolic statement. During the process of building a mathematical model, the modeler will decide what factors are relevant to the problem and what factors can be de-emphasized. Once a model has been developed and used to answer questions, it should be critically examined and often modified to obtain a more accurate reflection of the observed reality of that phenomenon. In this way, mathematical modeling is an evolving process; as new insight is gained, the process begins again as additional factors are considered. "Generally the success of a model depends on how easily it can be used and how accurate are its predictions." (Edwards & Hamson, 1994, p.3)

Building a Mathematical Model

Building a mathematical model for your project can be a challenging, yet interesting, task. A thorough understanding of the underlying scientific concepts is necessary and a mentor with expertise in your project topic is invaluable. It is also best to work as part of a team to provide more brainstorming power. In industry and engineering, it is common practice for a team of people to work together in building a model, with the individual team members bringing different areas of expertise to the project.

Although problems may require very different methods of solution, the following steps outline a general approach to the mathematical modeling process:

  1. Identify the problem, define the terms in your problem, and draw diagrams where appropriate.
  2. Begin with a simple model, stating the assumptions that you make as you focus on particular aspects of the phenomenon.
  3. Identify important variables and constants and determine how they relate to each other.
  4. Develop the equation(s) that express the relationships between the variables and constants.

Verifying and Refining a Model

Once the model has been developed and applied to the problem, your resulting model solution must be analyzed and interpreted with respect to the problem. The interpretations and conclusions should be checked for accuracy by answering the following questions: In answering these questions, you may need to modify your model. This refining process should continue until you obtain a model that agrees as closely as possible with the real world observations of the phenomenon that you have set out to model.

Variables and Parameters

Mathematical models typically contain three distinct types of quantities: output variables, input variables, and parameters (constants). Output variables give the model solution. The choice of what to specify as input variables and what to specify as parameters is somewhat arbitrary and often model dependent. Input variables characterize a single physical problem while parameters determine the context or setting of the physical problem. For example, in modeling the decay of a single radioactive material, the initial amount of material and the time interval allowed for decay could be input variables, while the decay constant for the material could be a parameter. The output variable for this model is the amount of material remaining after the specified time interval.

Continuous in Time vs. Discrete in Time Models

Mathematical models of time dependent processes can be split into two categories depending on how the time variable is to be treated. A continuous in time mathematical model is based on a set of equations which are valid for any value of the time variable. A discrete in time mathematical model is designed to provide information about the state of the physical system only at a selected set of times.

The solution of a continuous in time mathematical model provides information about the physical phenomenon over a continuum of time values. The solution of a discrete in time mathematical model provides information about the physical system at a finite number of time values. Continuous in time mathematical models have two advantages over discrete in time models: (1) they provide information at all times and (2) they more clearly show the qualitative effects which can be expected when a parameter or an input variable is changed. Discrete in time models, on the other hand, have two advantages over continuous in time models: (1) they are less demanding with respect to skill level in algebra, trigonometry, calculus, differential equations and (2) they are better suited for implementation on a computer. Most of the models that we present are discrete in time models.
 

Some Examples of Mathematical Models

A Falling Rock
A Spring Mass System
Heat Flow
Population Growth

Problem 1

Rotating all or part of a space station can create artificial gravity in the station. The resulting centrifugal force will be indistinguishable from gravitational force. Develop a mathematical model that will determine the rotational rate of the station as a function of the radius of the station (distance from the center of rotation) and the desired artificial gravitational force. Use this model to answer the question: What rotational rate is needed if the radius of the station is 150 m and Earth surface gravity is desired? Detailed solution.

Problem 2

A stretch of Interstate 25 is being widened to accommodate increasing traffic going north and south. Unfortunately, the Department of Transportation is going to have to bring out the orange barrels and close all but one lane at the "big I" intersection . The department would like to have traffic move along as quickly as possible without additional accidents. What speed limit would provide for maximum, but safe, traffic flow? Detailed solution.

More math modeling exercises.