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Math Modeling for Undergraduates

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Math Modeling for Undergraduates Project Number: MA-RYL-1213 Math Modeling for Undergraduates A Major Qualifying Project submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE in partial fulfillment of the requirements for the Degree of Bachelor of Science by Warren Anderson Phillip Blake December 14, 2012 Approved Professor Roger Y. Lui Major Advisor Abstract This project attempts to write the first draft of a mathematical modeling textbook for un- dergraduates. We based our information from the course we took under Professor Roger Lui in B term of 2011. The chapters in this book include: simple pendulum, escape velocity, Lotka-Volterra equations, extension on Lotka-Volterra equations, populations in competition, infectious diseases, chemostat, chemical kinetics, enzyme-substrate model, Poincar´e-Bendixson Theorem, and limit cycles. Emphasis of this book is on non-dimensionalization and phase plane analysis, which includes the study of the stability properties of the steady states. We also use the MATLAB program pplane8.m to help us plot the phase planes of various models. 1 Acknowledgments We thank our advisor, Professor Roger Lui, for his advice and help on this project. We also thank the students in the mathematical modeling class during B term of 2012 for their comments and feedback on our book. Finally, we thank Professor John Polking of Rice University who created the pplane8 MATLAB program, which was a tremendous help to us in plotting phase planes. 2 Contents 1 The Simple Pendulum Model 5 1.1 Introduction . .5 1.2 The Model . .5 1.3 The Equations . .7 1.4 Phase Plane Analysis with Direction Fields . 10 1.5 Conclusions . 12 1.6 Related Questions . 12 2 The Escape Velocity Problem 14 2.1 Introduction . 14 2.2 The Equations . 14 2.3 Solutions . 14 2.4 Conclusions and Questions . 16 3 The Lotka-Volterra Equations 17 3.1 Introduction . 17 3.2 The Lotka-Volterra Equations . 17 3.3 Steady States and their Stability . 19 3.4 Phase Plane Analysis . 22 3.5 Conclusion . 25 3.6 Problems . 25 4 Extension on the Lotka-Volterra Equations 27 4.1 Introduction . 27 4.2 A Logistic Modification of the Lotka-Volterra Equations . 27 4.3 Steady States and their Stability . 29 4.4 Phase Plane Analysis . 31 4.5 Conclusion . 35 4.6 Problems . 35 5 Populations in Competition 37 5.1 Introduction . 37 5.2 The Competition Model . 37 5.3 Nondimensionalization . 39 5.4 Steady States and their Stability . 41 3 5.5 Phase Plane Analysis . 42 5.6 Conclusion . 46 5.7 Problems . 47 6 Models for Infectious Diseases 48 6.1 Introduction . 48 6.2 Models for Infectious Diseases . 49 6.3 Steady Sates for the SIRS Model and their Stability . 50 6.4 Phase Plane Analysis . 52 6.5 Conclusion . 55 6.6 Problems . 55 7 The Chemostat Model 57 7.1 Introduction . 57 7.2 The Model . 58 7.3 The Equations . 58 7.4 Non-Dimensionalization . 61 7.5 Steady State Solutions . 63 7.6 Phase Plane Analysis and Stability . 64 7.7 Phase Plane Analysis with Direction Fields . 66 7.8 Related Questions . 67 8 Chemical Kinetics 68 8.1 Introduction . 68 8.2 General Chemical Equations and the Reaction Law . 68 8.3 Rate Law for the Formation of Water . 69 d[H2O] 8.4 The Graph of dt .................................. 72 8.4.1 Case 1: [H2]o > 2[O2]o ............................ 72 8.4.2 Case 2: [H2]o < 2[O2]o ............................ 73 8.5 Autocatalytic Reactions . 74 8.5.1 Example 1 . 75 8.5.2 Example 2 . 78 9 The Enzyme-Substrate Model 83 9.1 Introduction . 83 9.2 The Enzyme-Substrate Model . 83 9.3 Perturbation Method . 86 9.4 Problems . 89 10 Limit Cycles and the Poincar´e-BendixsonTheorem 91 10.1 Introduction . 91 10.2 Poincar´e-BendixsonTheorem . 92 10.3 Example 1: Scaled Predator-Prey Model . 93 10.4 Example 2: Converting to Polar Coordinates . 97 10.5 Example 3: The Van der Pol Equation . 102 4 Chapter 1 The Simple Pendulum Model 1.1 Introduction As we begin our study of differential equations we will start with a basic model, the simple pendulum. As with all the models that we will study, we can divide the study of the model into a few simple sections: Introduction, The Model, The Equations, Phase Plane Analysis with Direction Fields, and Conclusions. We will delve into what each of these sections entail as we come to them. The simple pendulum is defined as a pendulum that moves from a frictionless pivot with a massless rod connecting the mass to the pivot. If the rod were to have a non-negligible mass then it would have a different more complicated motion and therefore it would not be a simple pendulum. The same reasoning applies to the fact that this pendulum has an unbendable rod instead of some sort of rope or string. The length and shape of the rod are assumed to remain unchanged in all specific instances of the model. The mass can swing around the pivot point in a full 360o motion in two dimensional space. 1.2 The Model In order to derive the equatitons for our model, we will rely on the physical property of balanced forces. To begin we define a few variables. Let s = arc length which will denote the distance the pendulum is from its equilibrium. Let θ denote the positive angle in radians between the rod's current position and where the rod is positioned when the pendulum is at rest. Let L denote the length of the rod from the pivot point to the mass. Using these three properties, we can now develop a useful relationship that will help us later in developing our model equations. s θ = (1.1) 2πL 2π Or more simply that: s = θL (1.2) You can extend this to say that s¨ = θL¨ (1.3) 5 Figure 1.1: Free Body Diagram for Forces on Simple Pendulum . Now to build our equation we must examine the free body diagram of the simple pendulum. A free body diagram is a drawing which depicts the various forces acting on a system. In this case we are focusing on the forces which are causing the rod with its mass m to move about the pivot point. First and foremost we have that gravity is acting in the downward (negative) direction of the diagram. This force can be written as −mg where m is the mass of the weight at the end of the rod and g is the acceleration due to gravity. The tangential force T lays tangent to the arc length of the path of the pendulum. T can also be referred to as the restoring force which returns the rod back to its equilibrium position. We can relate these two forces with a force balance by forming a resultant vector between the two of them. We can express this relationship in terms of cosines. Remembering that θ is the angle between the resting position and the current position of the rod: π T cos − θ = (1.4) 2 −mg 6 We then rewrite this equation as : π T = −mg cos − θ (1.5) 2 Now using the property that: cos (a − b) = [cos a cos b − sin a sin b] We are able to rewrite the equation in such a way that h π π i T = −mg cos cos(θ) − sin sin(θ) (1.6) 2 2 π Now using properties of trigonometric functions it is apparent that cos( 2 ) = 0 and the fact π that sin( 2 ) = 1 we reduce our equation to: T = mg sin(θ) (1.7) Now to balance this tangential force as we proposed earlier we will use the properties of a force balance (more specifically Newton's 2nd Law F = ma) so find our model's equations. As we indicated earlier, in Equation (1.3), we saw that s¨ = θL¨ This equation can be rewritten as ms¨ = mθL¨ (1.8) Using Newton's 2nd Law F = ma, we can set Equation (1.8) equal to a force becauses ¨ represents acceleration. Adding this force to our already derived tangential force we will have a complete force balance that adds to zero. So our new, almost complete equation can be written as mLθ¨ + mg sin(θ) = 0 (1.9) Finally we can divide by m on both sides, the significance of this operation implies that the mass of a simple pendulum does not matter when describing it's motion mathematically. So our model's equation can be written as Lθ¨ + g sin(θ) = 0 (1.10) More concisely it can be written as g θ¨ + sin(θ) = 0 (1.11) L 1.3 The Equations To make our model more realistic, we could add air resistance into the equation. In physics air resistance is assumed to be proportional to velocity. Since we have already determined that 7 θ¨ is acceleration, we can safely say that θ_ is equivalent to velocity. So our new equation with resistance can be written as: g θ¨ + c θ_ + sin θ = 0 (1.12) L Here, c is the coefficient of friction which accounts for air resistance. From now on we will g 2 refer to L as !0. The reason this value is squared will prove to be useful shortly. From here we can attempt to solve our equation for θ(t). To begin we write our system of given equations which will include initial values θ0 and θ1 for our inital angle and velocity. We will first solve for the undamped case with no air resistance and we will then solve for the damped case with air resistance.
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