Mathematical models are a precise description of a problem, process, or technology in the form of mathematics. These models are built to learn more about a technology, system or method. The models explain why the system or process works the way it does and helps to study the effects and predict outcomes. Mathematical models are essential to scientists and engineers because it enables us to study phenomena that cannot be seen or measured. Models are also created by social scientists, economists, business analysts, and statisticians.
Example mathematical models are:
• The amount of heat that a fuel cell generates
• The amount of fuel needed for a fuel cell
• The amount of electricity generated by a fuel cell, battery or solar cell
• The amount of wind required to produce a certain amount of power by a wind tower
• The fluid flow rate through a fuel cell flow channel
• Dice roll outcomes
• Weather conditions such as tropical storms and hurricanes
• The stability of a roller coaster
• Stock market outcomes
The steps required for creating a mathematical model are comparable to the phases used for the scientific method. This includes observing, researching, building a model and predicting the outcome.
The observation phase includes observing and measuring what is happening in the actual world. This can consist of gathering empirical data or quantitative measurements. If we want to predict the amount of heat that a fuel cell generates, the observation stage may include of measuring the temperature of different parts of the fuel cell (such as the MEA, bipolar plates, end plates, and the temperature of the surroundings). We may notice that the MEA is heating up, which affects the other temperatures that are being measured. From this observation, we may choose to create a hypothesis. We may not understand at this point why the MEA is heating up.
The research phase consists of performing research via the internet, books, and scientific papers and publications. If we are still working to predict the amount of heat that a fuel cell generates, we may do research and find out that there are equations that include the temperature in the catalyst layer of the MEA.
The modeling phase includes creating or inputting the necessary equations into software such as Excel, MATLAB, Mathematica, SASS, or another software program. Many equations are often put into a model to predict an outcome. Empirical models are created using data gathered during the observation phase to generate one or more equations using regression or similar methodology. In our heat prediction example, we may put the polarization equations to create our model.
The prediction phase uses the model designed to predict what will happen in a yet-to-be-conducted experiment or an anticipated set of events in the real world. In our heat prediction example, we would use our model to predict the fuel cell and surrounding temperatures after a particular time (such as running continuously for 72 hours).
Most scientists and engineers conduct experiments rather than create models, which means that they collect observations and convert those observations into a mathematical model. The experimentalist designs a study, conducts experiments, and then records and analyzes the results. In contrast, the mathematical modeler will use some of the aspects of the real-world problem as inputs into their numerical system, but they come up with their equations from research or other studies.
Engineers often use models of varying complexity to design equipment, devices, and processes correctly. For example, every vehicle, airplane or building is the result of a model-based prediction that:
• The vehicle will operate as designed
• The building will stand
• The aircraft will fly
Prediction in engineering design assumes that resources can be invested with confidence because the predicted outcome will be correct. Depending upon the design and model, it may also eliminate months or years of experimental time. Experiments often only adequately measure and analyze a certain percent of the variables. Also, when investigating new technology, the importance of individual variables may be unknown or unable to be measured. For a mathematical model to be thorough, it must include all critical variables.
When you are thinking about creating a mathematical model, you already have some observations about the real world. You should be able to identify the need for the model easily:
• You cannot see the phenomena occurring directly
• The number of experiments would be too costly or time-consuming
• You are uncertain about what is occurring
When you are ready to create a mathematical model, some questions and thoughts that may be helpful are listed below:
1) What are we looking for? How should we look at this model? Identify the central principles. Begin with observations about the real world and gather the information that is relevant to the mathematical model.
2) What do we want to know? A list of questions should be created that needs to be answered. After you have decided on the scope of the problem, all relevant data should be identified. The questions “what do we know?” or “what information is available to help solve our problem?”
3) What do we know from experiments and literature? Conduct a thorough literature search. It may be possible that someone has already created a mathematical model of the process or problem that you are trying to solve.
4) How should we look at his model? Draw diagrams of what you want to model. This will help to clarify your model inputs and outputs and will begin to define the problem scope.
5) What assumptions can we make to eliminate variables? Create a list of the assumptions to clarify the scope of the model.
6) What will our model predict? Start with a simplistic model, and then add complexity as needed. Identify and construct the equations that will be used and the resulting answers. If you are building an empirical model, create equations from the data that you have obtained.
7) What are the input & output variables? Create a list of input and output variables. Define each constant and determine the variables that you need to solve.
8) Are the results valid? Validate your model with data that you did not use to create the model. Identify tests that can validate the model.
9) Continually test your model and update your equations based upon new data and information.
If there is good agreement between the observations and the model predictions, then the mathematical system does capture the essential aspects of the real-world scenario. However, some mathematical predictions may not agree closely with observed events. This is common when you are creating a new model or if you have gathered insufficient data. When this occurs, the model should be modified to improve its accuracy. The incorrect predictions may help you to rethink the assumptions of the mathematical model. The incorrect assumptions of the revised model will lead to a more sophisticated and accurate model version. The goal is not to necessarily create the most precise model of your problem or process, but that the model predicts an actual outcome.
There are many ways to create the same mathematical model! Contrary to what most beginning modelers may think, there is no “right” way. There are two main classifications for models -- empirical and non-empirical:
1) Empirical models are created from actual data and past experiments.
2) Non-empirical models use equations based on existing relations in literature. The models may also include new equations derived by the person conducting the mathematical modeling.
Mathematical models are often dynamic systems, but they can also be statistical or logical in nature:
1) Dynamic or static: Models that are dynamic use a realistic approach because they consider time or space. This is usually accomplished through differential equations (DEs) or partial differential equations (PDEs). Models that are static often only have a single position relative to time or space.
2) Deterministic or probabilistic: A deterministic model is where the model can predict each state. These models perform similarly for a specific set of conditions. In a probabilistic model, randomness must be accounted for by probability distributions.
3) Empirical: This model is based on experimental results. The data is statistically analyzed, and then empirical equations are developed based upon the data.
If your model is accurate, then it can be run with different inputs and conditions to predict outcomes. A robust mathematical model can replace experiments. If significant data has been collected, it can be analyzed for patterns to create models. The process of building models forces the scientist or engineer to carefully look at all aspects of the problem and reevaluate our beliefs. This process helps to improve our understanding of the problem to be solved. This is a more thorough approach than only collecting data. Scientists that do not build models sometimes find that after obtaining a significant amount of data, that they have gathered the wrong data.
This may sound intimidating for the beginner in mathematical modeling, but I would recommend just starting as simply as possible. This may involve using equations from textbooks or previous classes that you have taken to generate an answer that seems reasonable to you. This makes it easy to create a model, and complexity can be built into a model as required. Experienced modelers may start with more complexity because they understand how to code the model properly, and to fit it into their existing model.