Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

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Note: To read these pages properly, you will need the latest version of the Mozilla Firefox browser, with the STIX fonts installed. In a few sections, you will also need the latest Java plug-in, and JavaScript must be enabled. If you use a browser other than Firefox, you should be able to access the pages and run the applets. However, mathematical expressions will probably not display correctly. Firefox is currently the only browser that supports all of the open standards.

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Rating

Student: contains scenes of mild algebra or calculus that may require guidance.

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Section Starter Question

Do you believe in the ideal gas law? Does it make sense to “believe in” an equation? What do we really mean when we say we “believe in” an equation?

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Key Concepts

1. All mathematical models are wrong, but some mathematical models are useful.
2. If the modeling assumptions are satisfied, proper mathematical models should predict well given a wide range of conditions corresponding to the assumptions.
3. Knowing that all mathematical models are wrong, the real question becomes how wrong the model must be before it becomes unacceptable.
4. When observed outcomes deviate unacceptably from predicted ideal behavior in honest scientific or engineering work, then we must then alter our assumptions, re-derive the quantitative relationships, perhaps with more sophisticated mathematics or introducing more quantities and begin the cycle of modeling again.

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Vocabulary

1. A mathematical model is a mathematical structure (often an equation) expressing a relationship among a limited number of quantifiable elements from the “real world” or some isolated portion of it.

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Mathematical Ideas

Remember the following proverb: All mathematical models are wrong, but some mathematical models are useful.

Mathematical Modeling

Mathematical modeling involves two equally important activities:

• Building a mathematical structure, a model, based on hypotheses about relations among the quantities that describe the real world situation, and then deriving new relations,
• Evaluating the model, comparing the new relations with the real world and making predictions from the model.

Good mathematical modeling explains the hypotheses, the development of the model and its solutions, and then supports the findings by comparing them mathematically with the actual circumstances. Successful modeling requires a balance between so much complexity that making predictions from the model may be intractable and so little complexity that the predictions are unrealistic and useless. A successful model must allow a user to consider the effects of different policies.

At a more detailed level, mathematical modeling involves successive steps in the cycle of modeling:

1. Factors and observations,
2. Mathematical structure,
3. Testing and sensitivity analysis,
4. Effects and observations.

Consider the diagram in Figure 1 which illustrates the cycle of modeling. Steps 1 and 2 in the more detailed cycle are the first activity described above and steps 3 and 4 are the second activity.

Modeling

A good description of the model will begin with an organized and complete description of important factors and observations. The description will often use data gathered from observations of the problem. It will also include the statement of scientific laws and relations that apply to the important factors. From there, the model must summarize and condense the observations into a small set of hypotheses that capture the essence of the observations. The small set of hypotheses is a restatement of the problem, changing the problem from a descriptive, even colloquial, question into a precise formulation that moves the statement from the general to the specific. This sets the stage for the modeler to demonstrate a clear link between the listed assumptions and the building of the model.

The hypotheses translate into a mathematical structure that becomes the heart of the mathematical model. Many mathematical models, particularly those from physics and engineering, become a single equation but mathematical models need not be a single concise equation. Mathematical models may be a regression relation, either a linear regression, an exponential regression or a polynomial regression. The choice of regression model should explicitly follow from the hypotheses since the growth rate is an important consequence of the observations. The mathematical model may be a linear or nonlinear optimization model, consisting of an objective function and a set of constraints. Again the choice of linear or nonlinear functions for the objective and constraints should explicitly follow from the nature of the factors and the observations. For dynamic situations, the observations often involve some quantity and its rates of change. The hypotheses express some connection between these quantities and the mathematical model then becomes a differential equation, either linear or nonlinear depending on the explicit details of the scientific laws relating the factors considered. For discretely sampled data instead of continuous time expressions the model may become a difference equation. If an important feature of the observations and factors is noise or randomness, then the model may be a probability distribution or a stochastic process. The classical models from science and engineering usually take one of these classical equation-like forms but not all mathematical models need to follow this format. Models may be a connectivity graph, or a group of transformations.

If the number of variables is more than a few, or the relations are complicated to write in a concise mathematical expression then the model can be a computer program. Programs written in either high-level languages such as C, FORTRAN or Basic and very-high-level languages such as MATLAB or a computer algebra system are mathematical models. Spreadsheets combining the data and the calculations are a popular and efficient way to construct a mathematical model. The collection of calculations in the spreadsheet express the laws connecting the factors which are represented by the data in the rows and columns of the spreadsheet. Some mathematical models may be expressed by using more elaborate software specifically designed for modeling. Some software allows the user to describe the connections between factors graphically to create and alter a model.

Although this set of examples of mathematical models varies in theoretical sophistication and the equipment used, the core of each is to connect the data and the relations into a mechanism that allows the user to vary elements of the model. Creating a model, whether a single equation, a complicated mathematical structure, a quick spreadsheet, or a large program is the essence of the first step connecting the boxes labeled 1 and 2 above.

First models need not be sophisticated or detailed. For beginning analysis “back of the envelope calculations” and “dimensional analysis” will be as effective as spending time setting up an elaborate model or solving equations with advanced mathematics. Unit analysis to check consistency and outcomes of relations is important to check the harmony of the modeling assumptions. A good model pays attention to units, the quantities should be sensible and match. Even more important, a non-dimensionalized model reveals significant relationships, and major influences. Unit analysis is an important part of modeling, and goes far beyond simple checking to make sure “units cancel.” [3, 2]

Mathematical Solution

Once the modelers create the model, then they should derive some new relations among the important quantities selected to describe the real world situation. This is the step connecting the boxes labeled 2 and 3 in the diagram. If the model is an equation, for instance the Ideal Gas Law, then one can solve the equation for one of the variables in terms of the others. In the Ideal Gas Law, solving for one of the gas parameters is quite easy. A regression equation model may require almost no mathematical solution, although it might be useful to find auxiliary quantities such as rates of growth or maxima or minima. For an optimization problem the solution is the set of optima or the rates of change of optima with respect to the constraints. If the model is a differential equation or a difference equation, then the solution may have some mathematical substance. For instance, for a ballistics problem, the model may be a differential equation and the solution by calculus methods yields the equation of motion. For a problem with randomness, the derivation may find the mean or the variance. For a connectivity graph, one might be interested in the number of cycles, components or the diameter of the graph. If the model is a computer program, then this step usually involves running the program to obtain the output.

It is easy for students to focus most attention on this stage of the process, since the usual methods are the core of the typical mathematical curriculum. This step usually requires no interpretation, the model dictates the methods that must be used. This step is often the easiest in the sense that it is the clearest on how to proceed, although the mathematical procedures may be daunting.

Testing and Sensitivity

Once this step is done, the model is ready for testing and sensitivity analysis. This is the step that connects the boxes labeled 3 and 4. At the least, the modelers should try to verify, even with common sense, the results of the solution. Typically for a mathematical model, the previous step allows the modelers to produce some important or valuable quantity of the model. Modelers compare the results of the model with standard or common inputs with known quantities for the data or statement of the problem. This may be as easy as substituting into the derived equation, regression expression, or equation of motion. When running a computer model or program, this may involve sequences of program runs and related analysis. With any model, the results will probably not be exactly the same as the known data so interpretation or error analysis will be necessary. The interpretation requires judgment on the relative magnitudes of the quantities produced in light of the confidence in the exactness or applicability of the hypotheses.

Another important activity at this stage in the modeling process is the sensitivity analysis. The modelers should choose some critical feature of the model and then vary the parameter value that quantifies that feature. The results should be carefully compared to the real world and to the predicted values. If the results do not vary substantially, then perhaps the feature or parameter is not as critical as believed. This is important new information for the model. On the other hand, if a predicted or modeled value varies substantially in comparison to the parameter as it is slightly varied, then the accuracy of measurement of the critical parameter assumes new importance. In sensitivity analysis, just as in all modeling, this comparison of “varying substantially” should be measured with significant digits, relative magnitudes, and rates of change. Here is another area where expressing parameters in dimensionless groups is important [3]. In some areas of applied mathematics such as linear optimization and statistics, a side effect of the solution method is that it produces sensitivity parameters. In linear optimization, these are sometimes called the shadow prices and these additional solution values should be used whenever possible.

Interpretation and Refinement

Finally the modelers must take the results from the previous steps and use them to refine the interpretation and understanding of the real world situation. This interpretation step is represented in the diagram by connection between the boxes labeled 4 and 1, completing the cycle of modeling. For example, if the situation is modeling motion, then examining results may show that the predicted motion is faster than measured, or that the object does not travel as far as the model predicts. Then it may be that the model does not include the effects of friction, and so friction should be incorporated into a new model. At this step, the modeler has to be open and honest in assessing the strengths and weaknesses of the model. It also requires an improved understanding of the real world situation to include the correct new elements and hypotheses to correct the discrepancies in the results.

The step between stages 4 and 1 may suggest new processes, or experimental conditions to alter the model. If the problem suggests changes then those changes should be implemented and tested in another cycle in the modeling process.

A good summary of the modeling process is that it is an intense and structured application of “critical thinking”. Sophistication of mathematical techniques is not always necessary, the mathematics connecting steps 2 and 3 or potentially steps 3 and 4 may only be arithmetic. The key to good modeling is the critical thinking that occurs between steps 1 and 2, steps 3 and 4, and 4 and 1. If a model does not fit into this paradigm, it probably does not meet the criteria for a good model.

Good mathematical modeling, like good critical thinking, does not arise automatically or naturally. The craft of creating, solving, using, and interpreting a mathematical model must be practiced and developed. The structured approach to modeling helps distinguish the distinct steps, each requiring separate intellectual skills. It also provides a framework for developing and explaining a mathematical model.

A classification of mathematical models

The goal of mathematical modeling is to represent a real-world situation in a way which is a close approximation, even ideally exact, with a mathematical structure which is still solvable. This represents the interplay between the mathematics occurring between steps 1 and 2 and steps 2 and 3. Then we can classify models based on this interplay.

1. Exact models with exact solutions.
2. Exact models with approximate solutions.
3. Approximate models with exact solutions.
4. Approximate models with approximate solutions.

Exact models are uncommon because only rarely do we know physical representations well enough to claim to represent situations exactly. Exact representations typically occur when in discrete situations where we can enumerate all cases and represent them mathematically.

Approximate models are much more typical.

Example. Modeling the motion of a pendulum is a classic problem in applied mathematics. Galileo first modeled the pendulum mathematically and mathematicians and physicists still investigate more sophisticated models. Modeling the pendulum is considered in most books on applied mathematics and differential equations, for example [2]. An overview of modeling the pendulum appears in [5].

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Figure 2: Schematic diagram of a pendulum

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As a physically inclusive model of the pendulum, consider the general elastic pendulum shown in Figure 2. The pendulum is a heavy bob $B$ attached to a pivot point $P$ by means of an elastic spring. The pendulum has at least 5 degrees of freedom relative to its rest position. The pendulum can swing in the $x$-$y$-$z$ spatial dimensions. The pendulum can rotation torsionally around the axis of the spring $PB$. Finally, the pendulum can lengthen and shorten along the axis $PB$. Although we usually think of a pendulum as the short rigid rod in a clock, some pendula move in all these degrees of freedom, for example a huge construction crane.

We usually simplify the modeling with assumptions:

1. The pendulum is a rigid rod so that it does not lengthen or shorten, nor does it rotate torsionally.
2. All the mass is concentrated in the bob $B$, that is, the pendulum rod is massless.
3. The motion occurs only in one plane, so that there is only one degree of freedom, measured by the angle $\theta$.
4. There is no air resistance or friction in the bearing at $P$.
5. The only forces on the pendulum are at the contact point $P$ and the force of gravity due to the earth.
6. The pendulum is small relative to the earth so that gravity acts in the “down” direction only and is constant.

Then with standard physical modeling with Newton’s laws of motion, we obtain the differential equation for the pendulum angle $\theta$ with respect to vertical

$$ml{\theta }^{\prime \prime }+mgsin\theta =0,\theta \left(0\right)={\theta }_0,{\theta }^{\prime }\left(0\right)=v_0.$$

Note the application of the math modeling process moving from step 1 to step 2 with the identification of important variables, the hypotheses that simplify, and the application of physical laws. It is clear that already the differential equation describing the pendulum is an approximate model.

Nevertheless, this nonlinear differential equation cannot be solved in terms of elementary functions. It can be solved in terms of a class of higher transcendental functions known as elliptic integrals, but in practical terms that is equivalent to not being able to solve the equation analytically. There are two alternatives, either to solve the equation numerically or to reduce the equation further to a yet simpler form.

For small angles $sin\theta \approx \theta$ so if the pendulum does not have large oscillations the model becomes

$$ml{\theta }^{\prime \prime }+mg\theta =0,\theta \left(0\right)={\theta }_0,{\theta }^{\prime }\left(0\right)=v_0.$$

This differential equation is analytically solvable with standard methods yielding

$$\theta \left(t\right)=Acos\left(\omega t+\varphi \right)$$

for appropriate constants $A$, $\omega$, and $\varphi$ derived from the constants of the model. This is definitely an exact solution to an approximate model. The approximation of $sin\theta$ by $\theta$ is made specifically so that the differential equation is explicitly solvable in elementary functions. The solution to the differential equation is the process that leads from step 2 to step 3 in the modeling process. The analysis necessary to measure the accuracy of the exact solution compared to the physical motion is difficult. The measurement of the accuracy would be the process of moving from step 2 to step 3 in the cycle of modeling.

For a clock maker the accuracy of the solution is important. There are $86,400$ seconds in a day, so an error on the order of even 1 part in 10,000 for a physical pendulum oscillating once per second accumulates unacceptably. This error concerned physicists such as C. Huygens who created advanced pendula with a period independent of the amplitude. This illustrates step 3 to step 4 and the next modeling cycle.

Models of oscillation, including the pendulum, have been a central paradigm in applied mathematics for centuries. Here we use the pendulum model to illustrate the modeling process leading to an approximate model with an exact solution.

An example from physical chemistry

This section illustrates the cycle of mathematical modeling with a simple example from physical chemistry. This simple example provides a powerful analogy to the role of mathematical modeling in mathematical finance. The historical order of discovery is slightly modified to illustrate the idealized modeling cycle. Scientific progress rarely proceeds in simple order.

Scientists observed that diverse gases such as air, water vapor, hydrogen, and carbon dioxide all behave predictably and similarly. After many observations, scientists derived empirical relations such as Boyle’s law, and the law of Charles and Gay-Lussac about the gas. These laws express relations among the volume $V$, the pressure $P$, the amount $n$, and the temperature. $T$ of the gas.

In classical theoretical physics, we can define an ideal gas by making the following assumptions [1]:

1. A gas consists of particles called molecules which have mass, but essentially have no volume, so the molecules occupy a negligibly small fraction of the volume occupied by the gas.
2. The molecules can move in any direction with any speed.
3. The number of molecules is large.
4. No appreciable forces act on the molecules except during a collision.
5. The collisions between molecules and the container are elastic, and of negligible duration so both kinetic energy and momentum are conserved.
6. All molecules in the gas are identical.

From this limited set of assumptions about theoretical entities called molecules physicists can derive the equation of state for an ideal gas in terms of the 4 quantifiable elements of volume, pressure, amount, and temperature. The equation of state or ideal gas law is

where $R$ is a measured constant, called the universal gas constant. This gives a simple algebraic equation relating the 4 quantifiable elements describing a gas. The equation of state or ideal gas law predicts very well the properties of gases under the wide range of pressures, temperatures, masses and volumes commonly experienced in everyday life. The ideal gas law predicts with accuracy necessary for safety engineering the pressure and temperature in car tires and commercial gas cylinders. This level of prediction works even for gases we know do not satisfy the assumptions, such as air, which chemistry tells us is composed of several kinds of molecules which have volume and do not experience completely elastic collisions because of intermolecular forces. We know the mathematical model is wrong, but it is still useful.

Nevertheless, scientists soon discovered that the assumptions of an ideal gas predict that the difference in the constant-volume specific heat and the constant-pressure specific heat of gases should be the same for all gases, a prediction that scientists observe to be false. The simple ideal gas theory works well for monatomic gases, such as helium, but does not predict so well for more complex gases. This scientific observation now requires additional assumptions, specifically about the shape of the molecules in the gas. The derivation of the relationship for the observable in a gas is now more complex, requiring more mathematical techniques.

Moreover, under extreme conditions of low temperatures or high pressures, scientists observe new behaviors of gases. The gases condense into liquids, pressure on the walls drops and the gases no longer behave according to the relationship predicted by the ideal gas law. We cannot neglect these deviations from ideal behavior in accurate scientific or engineering work. We now have to admit that under these extreme circumstances we can no longer ignore the size of the molecules, which do occupy some appreciable volume. We also must admit that intermolecular forces must be considered. The two effects just described can be incorporated into a modified equation of state proposed by J.D. van der Waals in 1873. Van der Waals’ equation of state is:

The additional constants $a$ and $b$ represent the new elements of intermolecular attraction and volume effects respectively. If $a$ and $b$ are small because we are considering a monatomic gas under ordinary conditions, the Van der Waals equation of state can be well approximated by the ideal gas law. Otherwise we must use this more complicated relation for engineering our needs with gases.

Physicists now realize that because of the complex nature of the intermolecular forces, a real gas cannot be rigorously described by any simple equation of state. We can honestly say that the assumptions of the ideal gas are not correct, yet are sometimes useful. Likewise, the predictions of the van der Waals equation of state describe quite accurately the behavior of carbon dioxide gas in appropriate conditions. Yet for very low temperatures, carbon dioxide deviates from even these modified predictions because we know that the van der Waals model of the gas is wrong. Even this improved mathematical model is wrong, but it still is useful.

An example from mathematical finance

For modeling mathematical finance, we make a limited number of idealized assumptions about securities markets. We start from empirical observations of economists about supply and demand and the role of prices as a quantifiable element relating them. We ideally assume that

1. The market has a very large number of identical, rational traders.
2. All traders always have complete information about all assets they are trading.
3. Prices vary randomly with some continuous probability distribution.
4. Trading transactions take negligible time.
5. Trading transactions can be made in any amounts.

These assumptions are very similar to the assumptions about an ideal gas. From the assumptions we will be able to make some standard economic arguments to derive some interesting relationships about option prices. These relationships can help us manage risk, and speculate intelligently in typical markets. However, caution is necessary. In discussing the economic collapse of 2008-2009, blamed in part on the overuse or even abuse of mathematical models of risk, Valencia [6] says “Trying ever harder to capture risk in mathematical formulae can be counterproductive if such a degree of accuracy is intrinsically unobtainable.” If the dollar amounts get very large (so that rationality no longer holds!), or only a few traders are involved, or sharp jumps in prices occur, or the trades come too rapidly for information to spread effectively, we must proceed with caution. The observed financial outcomes may deviate from predicted ideal behavior in accurate scientific or economic work, or financial engineering.

We must then alter our assumptions, re-derive the quantitative relationships, perhaps with more sophisticated mathematics or introducing more quantities and begin the cycle of modeling again.

Sources

Some of the ideas about mathematical modeling are adapted from the article by Valencia [6] and the book When Genius Failed by Roger Lowenstein. The classification of mathematical models and solutions is adpated from Sangwin, [5]. The example of an exact model with an exact solution is from the 2009 American Mathematics Competition 8.

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Problems to Work for Understanding

1. A ladder stand with one end on the floor and the other against a wall. It slides along the floor and down the wall. If a cat is sitting in the middle of the ladder, what curve does it move along?

   Create and solve a mathematical model for this situation, enumerating explicitly the simplifying assumptions that you make to create and solve the model.

2. How many jelly beans fill a 1-liter jar?

   Create and solve a mathematical model for this situation, enumerating explicitly the simplifying assumptions that you make to create and solve the model.

3. How many flat toothpicks would fit on the surface of a sheet of poster board?

   Create and solve a mathematical model for this situation, enumerating explicitly the simplifying assumptions that you make to create and solve the model.

4. If your life earnings were doled out to you at a certain rate per hour for every hour of your life, how much is your time worth?

   Create and solve a mathematical model for this situation, enumerating explicitly the simplifying assumptions that you make to create and solve the model.

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Reading Suggestion:

References

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Outside Readings and Links:

1. Duke University Modeling Contest Team. Accessed August 29, 2009

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Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1

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