MATHEMATICAL MODELING A Comprehensive Introduction
MATHEMATICAL MODELINGA Comprehensive IntroductionGerhard Dangelmayr and Michael KirbyDepartment of MathematicsColorado State UniversityFort Collins, Colorado, 80523PRENTICE HALL, Upper Saddle River, New Jersey 07458
ContentsPreface51 Mathematical Modeling1.1 Examples of Modeling . . . . . . . . . . . . . . . . . .1.1.1 Modeling with Difference Equations . . . . . .1.1.2 Modeling with Ordinary Differential Equations1.1.3 Modeling with Partial Differential Equation . .1.1.4 Optimization . . . . . . . . . . . . . . . . . . .1.1.5 Modeling with Simulations . . . . . . . . . . .1.1.6 Function Fitting: Data Modeling . . . . . . . .1.2 The Modeling Process . . . . . . . . . . . . . . . . . .1.2.1 An Algorithm for Modeling? . . . . . . . . . .1.3 The Delicate Science of Errors . . . . . . . . . . . . .1.4 Purpose of this Course . . . . . . . . . . . . . . . . . .7777889991010112 Qualitative Modeling with Functions2.1 Modeling Species Propagation . . . . . . . .2.2 Supply and Demand . . . . . . . . . . . . .2.2.1 Market Equilibrium . . . . . . . . .2.2.2 Market Adjustment . . . . . . . . .2.2.3 Taxation . . . . . . . . . . . . . . .2.3 Modeling with Proportion and Scale . . . .2.3.1 Proportion . . . . . . . . . . . . . .2.3.2 Scale . . . . . . . . . . . . . . . . . .2.4 Dimensional Analysis . . . . . . . . . . . . .2.4.1 Dimensional homogeneity . . . . . .2.4.2 Discovering Joint Proportions . . . .2.4.3 Procedure for Nondimensionalization2.4.4 Modeling with Dimensional 3 Linear Programming3.1 Examples of Linear Programs . . . . . . . .3.1.1 Red or White? . . . . . . . . . . . .3.1.2 How Many Fish? . . . . . . . . . . .3.2 Geometric Solution of a 2D Linear Program3.3 Sensitivity Analysis . . . . . . . . . . . . . .3.3.1 Price Sensitivity . . . . . . . . . . .3.3.2 Resource Sensitivity . . . . . . . . .3.3.3 Constraint Coefficient Sensitivity . .3.4 Linear Programs with Equality Constraints3.4.1 A Task Scheduling Problem . . . . .238.3940404141434344454546
33.53.63.4.2 Transportation Problems . . . . . . . . . . . . . . . . .A Targeting Problem . . . . . . . . . . . . . . . . . . . . . . . .3.5.1 Discretization and Solution of the Equations of Motion3.5.2 Formulation as Linear Program . . . . . . . . . . . . . .3.5.3 Targeting Problem with Air Resistance . . . . . . . . .3.5.4 Additional Constraints . . . . . . . . . . . . . . . . . . .Analysis of the Targeting Problem . . . . . . . . . . . . . . . .3.6.1 Analytical Solution . . . . . . . . . . . . . . . . . . . . .3.6.2 Dimensionless Variables . . . . . . . . . . . . . . . . . .3.6.3 Maximum Altitude . . . . . . . . . . . . . . . . . . . . 94 Modeling with Nonlinear Programming4.1 Unconstrained Optimization in One Dimension . . .4.1.1 Bisection Algorithm . . . . . . . . . . . . . .4.1.2 Newton’s Method . . . . . . . . . . . . . . . .4.2 Unconstrained Optimization in Higher Dimensions .4.2.1 Taylor Series in Higher Dimensions . . . . . .4.2.2 Roots of a Nonlinear System . . . . . . . . .4.2.3 Newton’s Method . . . . . . . . . . . . . . . .4.2.4 Steepest Descent . . . . . . . . . . . . . . . .4.3 Constrained Optimization and Lagrange Multipliers4.4 Geometry of Constrained Optimization . . . . . . . .4.4.1 One Equality Constraint . . . . . . . . . . . .4.4.2 Several Equality Constraints . . . . . . . . .4.4.3 Inequality Constraints . . . . . . . . . . . . .4.5 Modeling Examples . . . . . . . . . . . . . . . . . . .5 Empirical Modeling with Data Fitting5.1 Linear Least Squares . . . . . . . . . . . . . .5.1.1 The Mammalian Heart Revisited . . .5.1.2 General Formulation . . . . . . . . . .5.1.3 Exponential Fits . . . . . . . . . . . .5.1.4 Fitting Data with Polynomials . . . .5.1.5 Interpolation versus Least Squares . .5.2 Splines . . . . . . . . . . . . . . . . . . . . . .5.2.1 Linear Splines . . . . . . . . . . . . . .5.2.2 Cubic Splines . . . . . . . . . . . . . .5.3 Data Fitting and the Uniform Approximation5.3.1 Error Model Selection? . . . . . . . . .88. 89. 89. 89. 92. 92. 95. 96. 96. 97. 99. 1026 Modeling with Discrete Dynamical Systems6.1 Introduction . . . . . . . . . . . . . . . . . . .6.2 Linear First Order Difference Equations . . .6.2.1 Analytical Solutions . . . . . . . . . .6.2.2 Modeling Examples . . . . . . . . . .6.3 Linear Second Order Equations . . . . . . . .6.3.1 Homogeneous Equations . . . . . . . .106106110110115119119
46.46.56.3.2 The Cobweb Model Revisited . . . . . . . . . . . . . . . . . .Nonlinear Difference Equations and Systems in Population Modeling6.4.1 Systems of Equations and Competing Species . . . . . . . . .Empirical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.5.1 Non-Newtonian Fish? . . . . . . . . . . . . . . . . . . . . . .6.5.2 Predator or Prey? . . . . . . . . . . . . . . . . . . . . . . . .1221241251281281317 Simulation Modeling1407.1 The Tire Distributor Problem . . . . . . . . . . . . . . . . . . . . . . 1407.2 Blackjack Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142APPENDICESA Matlab Code for Data FittingA.1 Mammalian Heart Rate Problem . . . . . .A.2 Least Squares with Normal Equations . . .A.3 Least Squares with Overdetermined SystemA.4 Non-Newtonian Fish . . . . . . . . . . . . .A.5 Preditor or Prey? . . . . . . . . . . . . . . .A.6 Tire Distributor . . . . . . . . . . . . . . . .A.7 Blackjack . . . . . . . . . . . . . . . . . . .149149151153154154155158
PrefaceThese materials are being developed with support from National Science Foundation Award no. 0126650 entitled A Mathematical Modeling Program for Undergraduates in Science, Mathematics, Engineering and Technology.The objective of this project is the development of innovative educationalmaterials that incorporate a novel educational approach and perspective to enhancethe teaching and learning of mathematics for the purposes of knowledge discovery.The general undergraduate educated with these materials will possess a readilyapplicable toolbox of mathematical ideas for quantifying real world problems aswell as problem solving skills, and possibly the most importantly, the ability tointerpret results and further understanding.Our pedagogical perspective consists of the observation that mathematicalmodeling is often taught backwards. An application of interest is presented andthen appropriate mathematical tools are subsequently invoked. The beginner isleft with the obvious concern. How does one decide which method to use on a newproblem? Our proposed solution to this dilemma is to teach mathematics first andthen show why a given mathematical methodology can be applied to the modelingproblem. We will be successful if the student completes their modeling course basedon these materials with a good sense of what makes various mathematical methodsinherently different. Furthermore, students that are aware of the fundamental distinguishing characteristics of the array of methodologies should now be equipped toaddress this question of central importance in modeling, i.e., which method when!This text is the first of two planned works to establish ”proof of concept” ofa new approach to teaching mathematical modeling. The scope of the text is thebasic theory of modeling from a mathematical perspective. A second applicationsfocussed text will build on the basic material of the first volume.It is typical that students in a mathematical modeling class come from a widevariety of disciplines. In addition, their preparation and mathematical sophistication can vary as widely as their areas of interest. This heterogeneity makes theteaching and learning of mathematical modeling a significant challenge. One of themain student prototypes is a intelligent although possibly mathematically naivestudent that must learn mathematically modeling to make progress in an area ofresearch. If a course or textbook does not provide the necessary information forthese good students to bridge educational gaps students everyone suffers. Indeed,most textbooks fail to be accessible to such audiences.With enhancing accessibility as our motivation, we propose to implement asimple pedagogical device to facilitate the use of the text by students of widelyvarying backgrounds. This device consists of graded levels of presentation denotedby (E) for elementary, (I) for intermediate and (A) for advanced. (E) Mathematical beginners will find much of interest in the elementary sections as well as foundation material for further study. The diligent studentcan use this self-contained treatment to pave the way to reading of moreadvanced sections. The basic properties of mathematical techniques will bepresented with an emphasis on how methods lead to specific applications.5
6Preface (I) Intermediate material builds on the elementary material and extends thestudents expertise. Often intermediate material will involve computer experiments to stimulate more theoretical discussions in the advanced material.A good understanding of intermediate material should permit a student todevelop new applications of central mathematical ideas. (A) Advanced material will provide mathematically mature students with asolid theoretical foundation for the subject. Mastery of this subject matter should provide the student with the insight required to further developmathematical models.If a section is labeled as (E) then all its subsections are at the same level. If itis not labelled, then each individual subsection will be labelled for level of difficulty.These texts will be pilot tested at Colorado State University during the courseof development and will incorporate a fundamentally new approach to modelingthrough general mathematical principles rather than ad hoc lists of methods andtechniques. These methods will be demonstrated within the context of on-goingstate-of-the-art interdisciplinary research projects. (Such an approach will havethe added advantage of broadening students perspectives and appreciation for thenature of basic university research.) The basic aim of the materials is to present aninnovative approach to inform and educate students about the power and importance of basic mathematics and mathematical modeling in the process of knowledgediscovery.Michael KirbyGerhard Dangelmayr
C H A P T E R1Mathematical ModelingMathematical modeling is becoming an increasingly important subject as computers expand our ability to translate mathematical equations and formulations intoconcrete conclusions concerning the world, both natural and artificial, that we livein.1.1EXAMPLES OF MODELINGHere we do a quick tour of several examples of the mathematical process. Wepresent the models as finished results as opposed to attempting to develop themodels.1.1.1Modeling with Difference EquationsConsider the situation in which a variable changes in discrete time steps. If thecurrent value of the variable is an then the predicted value of the variable will bean 1 . A mathematical model for the evolution of the (still unspecified) quantityan could take the forman 1 αan βIn words, the new value is a scalar multiple of the old value offset by some constantβ. This model is common, e.g., it is used for modeling bank loans. One mightamend the model to make the dependence depend on more terms and to includethe possibility that every iteration the offset can change, thus,an 1 α1 an α2 a2n βnThis could correpsond to, for example, a population model where the the migrationlevels change every time step. In some instances, it is clear that information requiredto predict a new value goes back further than the current value, e.g.,an 1 an an 1Note now that two initial values are required to evolve this model. Finally, it maybe that the form of the difference equations are unknown and the model must bewrittenan 1 f (an , an 1 , an M 1 )Determining the nature of f and the step M is at the heart of model formulationwith difference equations. Often observed data can be employed to assist in thiseffort.1.1.2Modeling with Ordinary Differential EquationsAlthough modeling with ordinary differential equations shares many of the ideas ofmodeling with the difference equations discussed above, there are many fundamen7
8Chapter 1Mathematical Modelingtal differences. At the center of these differences is the assumption that time is acontinuous variable.One of the simplest differential equations is also an extremely importantmodel, i.e.,dx αxdtIn words, the rate of change of the quantity x depends on the amount of thequantity. If α 0 then we have exponential growth. If α 0 the situation isexponetial decay. Of course additional terms can be added that fundamentallyalter the evolution of x(t). For exampledx α1 x α2 x2dtThe model formulation again requires the development of the appropriate righthand side.In the above model the value x on the right hand side is implicitly assumed tobe evaluated at the time t. It may be that there is evidence that the instantaneousrate of change at time t is actually a function of a previous time, i.e.,dx f (x(t)) g(x(t
Mathematical modeling is becoming an increasingly important subject as comput-ers expand our ability to translate mathematical equations and formulations into concrete conclusions concerning the world, both natural and artificial, that we live in. 1.1 EXAMPLES OF MODELING Here we do a quick tour of several examples of the mathematical process. We
So, I say mathematical modeling is a way of life. Keyword: Mathematical modelling, Mathematical thinking style, Applied 1. Introduction: Applied Mathematical modeling welcomes contributions on research related to the mathematical modeling of e
2.1 Mathematical modeling In mathematical modeling, students elicit a mathematical solution for a problem that is formulated in mathematical terms but is embedded within meaningful, real-world context (Damlamian et al., 2013). Mathematical model
14 D Unit 5.1 Geometric Relationships - Forms and Shapes 15 C Unit 6.4 Modeling - Mathematical 16 B Unit 6.5 Modeling - Computer 17 A Unit 6.1 Modeling - Conceptual 18 D Unit 6.5 Modeling - Computer 19 C Unit 6.5 Modeling - Computer 20 B Unit 6.1 Modeling - Conceptual 21 D Unit 6.3 Modeling - Physical 22 A Unit 6.5 Modeling - Computer
What is mathematical modeling? – Modeling (Am -English spelling) or modelling (Eng spelling) is a mathematical process in which mathematical problem solvers create a solution to a problem, in an attempt to make sense of mathematical phenomena (e.g., a data set, a graph, a diagram, a c
Mathematical modeling is the process of using mathematical tools and methods to ask and answer questions about real-world situations (Abrams 2012). Modeling will look different at each grade level, and success with modeling is based on students’ mathematical background know
The need to develop a mathematical model begins with specific questions in a particular application area that the solution of the mathematical model will answer. Often the mathematical model developed is a mathematical “find” problem such as a scalar equation, a system o
cardiovascular modeling at the systems-physiology level. 1 Introduction Mathematical modeling has a long and very rich history in physiology. Otto Frank’s mathematical analysis of the arterial pulse, for example, dates back to the late 19th century [12]. Similar mathematical approaches to understanding the mechanical
ALBERT WOODFOX . CIVIL ACTION NO. 06-789-JJB-RLB . VERSUS . BURL CAIN, WARDEN OF THE LOUISIANA . STATE PENITENTIARY, ET AL. RULING . Before this Court is the pending Motion (doc. 279) for Rule 23(c) release of Petitioner, Albert Woodfox. Briefs were filed in response to this motion and were considered by this Court. Subsequently, a motion hearing on this matter was held before this Court on .
