What is Mathematical Modeling?
Mathematical models are a depiction of a problem, process, or technology in the form of mathematics. Models are built to learn about a system or process, study the affects, and predict outcomes. They explain why the system or process works the way it does, and why the results occurred. They allow us to study and predict results and phenomena that cannot be seen or cannot be measured. Mathematical models are created and used by those in natural science and engineering, social sciences, economists, business, and statisticians.
The process required for creating a mathematical model is the same steps used for the scientific method. This includes observing, creating a model, and then predicting the outcome. The observation stage includes observing and measuring what is occurring in the real world. This can consist of gathering empirical evidence or quantitative measurements. The modeling phase includes creating the necessary equations, and analyzing the observations noted thus far. The prediction part of the scientific method uses the model created to predict what will happen in a yet-to-be-conducted experiment or in an anticipated set of events in the real world.
Most scientists and engineers use the experimental approach rather than the modeling approach. The experimentalist designs a study, conducts the experiment, and then records and analyzes the results. In contrast, the modeler converts some of the aspects of the real-world problem into a mathematical system.
Engineers in particular uses models of varying complexity to predict process, device, or technology results in order to properly design devices and processes. For example, every vehicle, airplane or building represents a model-based prediction that the vehicle will operate correctly, the building will stand, or the airplane will fly without unanticipated consequences. Prediction in engineering design assumes that resources of time, imagination, and money can be invested with confidence because the predicted outcome will be a good one. Depending upon the design and the model, it may also eliminate months or years of experimental time. In some cases, experiments only adequately measure and analyze a certain percent of the variables. Also, when investigating a new technology, the importance of certain variables may not be known or are unable to be measured. In order for a mathematical model to be adequate, it must include and account for all of the important variables.
Steps for Creating a Mathematical Model
Usually when you begin thinking about creating a mathematical model, you have some observations about the real world. You should be able to easily identify the need for the model. When you are ready to being creating a mathematical model, some questions and/or thoughts that may be helpful are listed below:
1) What are we looking for? How should we look at this model?
Identify the governing principles. You usually begin with some observations about the real world, and gather all the information that you currently have that is relevant for the mathematical model.
2) What do we want to know?
A list should be created regarding the questions that need to be answered. After you have decided on the initial scope of the problem, all available relevant data should be identified. The question “what do we know?” or “what information is available to help solve the question of what we are looking to answer?
3) What do we already know from experiments and/or literature?
If you have not done so already, conduct a thorough literature search. It may be possible that someone already created a mathematical model of the process or problem that you are trying to solve.
4) How should we look at his model?
Create as many diagrams of what is actually happening with the process that you are trying to model. This will help to clarify your inputs and outputs, and will begin to clarify the define the scope around the problem that you are trying to solve.
5) What assumptions can we make to eliminate some of the variables?
Create a list of all of the assumptions that you will use to narrow and clarify the scope of the model.
6) What will our model predict? Start with a simple model, and then add complexity as needed. Identify and/or construct the equations that will be used, and the resulting answers. If you are building an empirical model, then create equations from the data that you have obtained.
7) What are the input & output variables?
Create a list of all of your input and output variables. Define each constant, and determine which variables that you need to solve for, and others that you do not have constants for.
8) Are the results valid?
Validate your model with new experimental data or data that you have not used to create the model. Identify tests that can validate the model.
9) Constantly test your model and update your equations based upon new data and information.
If there is good agreement between what is observed and what the model predicts, then there is some reason to believe that the mathematical system does indeed capture correctly important aspects of the real-world situation. However, some of the predictions of a mathematical model may agree closely with observed events, while others may not. When this occurs, the model need to be modified to improve its accuracy. The incorrect predictions may suggest ways of rethinking the assumptions of the mathematical system. The incorrect inferences of the revised model will lead, in turn, to yet another version, more sophisticated more accurate than the previous one. The goal is not to make the most precise model of your problem or process, but that model includes all the essential variables in order to accurately predict an outcome.
This may sound intimidating for the beginner in mathematical modeling, however, if you are new to creating mathematical models – I recommend starting as simply as possible. This may include using equations from textbooks or previous classes that you have taken, and create an answer that seems reasonable to you. This makes it easy to start creating a model, and complexity can be built into a model as needed. Experienced modelers may begin with more complexity because they understand how to code it properly, and to fit it in with the rest of their model.