WhatIsMathematical Modeling?

1 1 What Is MathematicalModeling?We begin this book with a dictionary definition of the wordmodel:model(n): a miniature representation of something; a pattern of some-thing to be made; an example for imitation or emulation; a description oranalogy used to help visualize something ( , an atom) that cannot be dir-ectly observed; a system of postulates, data and inferences presented as amathematical description of an entity or state of affairsThis definition suggests thatmodelingis an activity, acognitive activityinwhich we think about and make models to describe how devices or objectsof interest are many ways in which devices and behaviors can be can use words, drawings or sketches, physical models, computer pro-grams, or mathematical formulas. In other words, the modeling activitycan be done in several languages, often simultaneously. Since we are par-ticularly interested in using the language of mathematics to make models,34 chapter 1 What Is Mathematical Modeling?

2 We will refine the definition just given:mathematical model(n): a representation in mathematical terms of thebehavior of real devices and objectsWe want to know how to make or generate mathematical representationsor models, how to validate them, how to use them, and how and when theiruse is limited. But before delving into these important issues, it is worthtalking about why we do mathematical Wh yDo We Do Mathematical Modeling? Since the modeling of devices and phenomena is essential to both engi-neering and science, engineers and scientists have very practical reasonsfor doing mathematical modeling. In addition, engineers, scientists, andmathematicians want to experience the sheer joy of formulating and solvingmathematical Mathematical Modeling and theScientific MethodIn an elementary picture of thescientific method(see Figure ), we identifya real world and a conceptual world. The external world is the onewe call real; here we observe various phenomena and behaviors, whethernatural in origin or produced by artifacts.

3 The conceptual world is theworld of the mind where we live when we try to understand what isgoing on in that real, external world. The conceptual world can be viewedas having three stages: observation, modeling, and theobservationpart of the scientific method we measure what ishappening in the real world. Here we gather empirical evidence and factson the ground. Observations may be direct, as when we use our senses, orindirect, in which case some measurements are taken to indicate throughsome other reading that an event has taken place. For example, we oftenknow a chemical reaction has taken place only by measuring the productof that this elementary view of how science is done, themodelingpart isconcerned with analyzing the above observations for one of (at least) threereasons. These rationales are about developing:models that Wh yDo We Do Mathematical Modeling? 5 The real worldThe conceptual worldPhenomenaPredictionsObservationsMod els (analyses)Figure An elementarydepiction o fthescientificmethodthat shows how our conceptualmodels o fthe world are related toobservations made within that realworld (Dym and Ivey, 1980).

4 Behavior or results observed;models that explain whythat behavior andresults occurred as they did; ormodels that allow us to predictfuturebehaviors or results that are as yet unseen or thepredictionpart of the scientific method we exercise our modelsto tell us what will happen in a yet-to-be-conducted experiment or inan anticipated set of events in the real world. These predictions are thenfollowed by observations that serve either to validate the model or to suggestreasons that the model is last point clearly points to the looping, iterative structure apparentin Figure It also suggests that modeling is central to all of the conceptualphases in the elementary model of the scientific method. We build modelsand use them to predict events that can confirm or deny the models. Inaddition, we can also improve our gathering of empirical data when we usea model to obtain guidance about where to Mathematical Modeling and thePractice of EngineeringEngineers are interested indesigningdevices and processes and is, beyond observing how the world works, engineers are interestedin creating artifacts that have not yet come to life.

5 As noted by HerbertA. Simon (inThe Sciences of the Artificial), Design is the distinguishingactivity of engineering. Thus, engineers must be able to describe andanalyze objects and devices into order to predict their behavior to see if6 chapter 1 What Is Mathematical Modeling? that behavior is what the engineers want. In short, engineers need to modeldevices and processes if they are going to design those devices and the scientific method and engineering design have much in com-mon, there are differences in motivation and approach that are worthmentioning. In the practices of science and of engineering design, mod-els are often applied to predict what will happen in a future situation. Inengineering design, however, the predictions are used in ways that havefar different consequences than simply anticipating the outcome of anexperiment. Every new building or airplane, for example, represents amodel-based prediction that the building will stand or the airplane will flywithout dire, unanticipated consequences.

6 Thus, beyond simply validat-ing a model, prediction in engineering design assumes that resources oftime, imagination, and money can be invested with confidence because thepredicted outcome will be a good Principles of Mathematical ModelingMathematical modeling is aprincipledactivity that has both principlesbehind it and methods that can be successfully applied. The principles areover-arching ormeta-principles phrased as questions about the intentionsand purposes of mathematical modeling. These meta-principles are almostphilosophical in nature. We will now outline the principles, and in the nextsection we will briefly review some of the visual portrayal of the basic philosophical approach is shown inFigure These methodological modeling principles are also capturedin the following list of questions and answers: Why?What are we looking for? Identify the need for the model. Find?What do we want to know? List the data we are seeking.

7 Given?What do we know? Identify the available relevant data. Assume?What can we assume? Identify the circumstances that apply. How?How should we look at this model? Identify the governingphysical principles. Predict?What will our model predict? Identify the equations that willbe used, the calculations that will be made, and the answers that willresult. Valid?Are the predictions valid? Identify tests that can be madetovalidatethe model, , is it consistent with its principles andassumptions? Verified?Are the predictions good? Identify tests that can be madetoverifythe model, , is it useful in terms of the initial reason itwas done? Principles of Mathematical Modeling7 Why? What are we looking for?Find? What do we want to know?How? How should we look at this model?Given? What do we know? Assume? What can we assume?Predict? What will our model predict?Valid? Are the predictions valid?

8 Improve? How can we improve the model?Use? How will we exercise the model?OBJECT/SYSTEMMODEL VARIABLES, PARAMETERSV erified? Are the predictions good?MODEL PREDICTIONSVALID, ACCEPTED PREDICTIONSTESTF igure A first-order view ofmathematicalmodelingthatshows how the questions asked in a principled approach to buildinga model relate to the development o fthat model (inspired byCarson and Cobelli, 2001). Improve?Can we improve the model? Identify parameter values thatare not adequately known, variables that should have been included,and/or assumptions/restrictions that could be lifted. Implement theiterative loop that we can call model-validate-verify-improve-predict. Use?How will we exercise the model? What will we do with the model?This list of questions and instructions isnotan algorithm for buildinga good mathematical model. However, the underlying ideas are key tomathematical modeling, as they are key to problem formulation , we should expect the individual questions to recur often during themodeling process, and we should regard this list as a fairly general approachtoways of thinkingabout mathematical a clear picture of why the model is wanted or needed is of primeimportance to the model-building enterprise.

9 Suppose we want to estimatehow much power could be generated by a dam on a large river, say a damlocated at The Three Gorges on the Yangtze River in Hubei Province in thePeople s Republic of China. For a first estimate of the available power, we8 chapter 1 What Is Mathematical Modeling? wouldn t need to model the dam s thickness or the strength of its founda-tion. Its height, on the other hand, would be an essential parameter of apower model, as would some model and estimates of river flow , on the other hand, we want to design the actual dam, we would needa model that incorporates all of the dam s physical characteristics ( ,dimensions, materials, foundations) and relates them to the dam site andthe river flow conditions. Thus, defining the task is the first essential stepin model then should list what we know for example, river flow quantitiesand desired power levels as a basis for listing the variables or parametersthat are as yet unknown.

10 We should also list any relevant example, levels of desired power may be linked to demographic oreconomic data, so any assumptions made about population and economicgrowth should be spelled out. Assumptions about the consistency of riverflows and the statistics of flooding should also be spelled physical principles apply to this model? The mass of the river swater must be conserved, as must its momentum, as the river flows, andenergy is both dissipated and redirected as water is allowed to flow throughturbines in the dam (and hopefully not spill over the top!). And mass mustbe conserved, within some undefined system boundary, because dams doaccumulate water mass from flowing rivers. There are well-known equa-tions that correspond to these physical principles. They could be usedto develop an estimate of dam height as a function of power can validate the model by ensuring that our equations and calcu-lated results have the proper dimensions, and we can exercise the modelagainst data from existing hydroelectric dams to get empirical data we find that our model is inadequate or that it fails in some way, wethen enter aniterative loopin which we cycle back to an earlier stage of themodel building and re-examine our assumptions, our known parametervalues, the principles chosen, the equations used, the means of calculation,and so on.