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Mathematical modeling is an important topic in today’s information age. Knowledge of ways to analytically represent and study complex problems in the physical and social sciences is in high demand. Introducing students to methods of mathematical modeling will acquaint them with these problems and enable them to consider graduate study or careers in the applied analytical fields.
MT470 Mathematical Modeling: 3 Credit Hours
Prerequisites: MT202 Multivariable Calculus, MT210 Linear Algebra.
This course introduces ways to represent and study today’s complex modeling problems, with cases from the natural and social sciences. The study of finite difference (recurrence) equations as models of discrete dynamical systems serves as the core mathematical theory taught in this course. Students are shown how linear algebra methods are used solve models expressed as linear dynamical systems. The course then deep dives into ways to understand the dynamic, asymptotic, and chaotic behaviors of nonlinear discrete models. This includes the study of fixed points, periodic points, orbits, and stability theorems of 1-, 2-, and 3-dimensional maps. Cases studied with these attributes include the dynamics of species models and the structure of epidemiological models of pathogen transmission in human populations. Deterministic and Markov chain models of these cases will be presented.
Required Texts
Samuel Goldberg, Introduction to Difference Equations, Dover Publications, New York.
Glenn Fulford, Peter Forrester, Arthur Jones, Modelling with Differential and Difference Equations, Cambridge University Press, 1997.
Ronald E. Mickens, Difference Equations: Theory, Applications, and Advanced Topics, 3rd Edition, CRC Press, 2018 (paperback version).
Classroom, Office Hours, and Contact Info
Classroom and Meeting Times: Monday, Wednesday, 4:30-5:45 pm, Gasson Hall, Room 210.
Office & Office Hours: Maloney Hall, Office 517, Office Hours: By Appointment and Zoom Scheduled.
Exams and Course Grade
Problem Sets: There are two types of problem sets – graded and ungraded.
Ungraded problem sets and their solutions are posted on Canvas. Ungraded problem sets are not turned in; however, they are essential to do for the exams. Graded problem sets are those given throughout the semester as the lectures proceed. Graded problem sets are turned in for a grade and they will be a factor in your semester grade. Graded problem sets are to be worked on individually.
Mid Term Exams: Wednesday 6 October. Wednesday 17 November. Each midterm is 1/3 of the course grade. Mid-term exams are subject to slight changes depending on progress through the lectures.
Last Lecture Before Final Exam: Monday 8 December, Reading Period: 10-13 December.
Final Exam: Monday 20 December, Gasson 210, 4:30-6:30 pm. The final is 1/3 of the course grade.
Calculators and Microsoft Excel
Calculators are permitted. Ability to use Microsoft Excel is required. Excel Macro or Visual Basic programming is nice but not needed. Familiarity with MATLAB or Mathematica is nice but not needed.
CANVAS Course Site
A CANVAS site has been published for the course.
Academic Integrity
Better to honorably fail than to dishonorably achieve – the latter being the true failure. Read BC’s policy at www.bc.edu/integrity. Academic dishonesty has serious consequences.
ath 470 – Applied Mathematics I – Course Outline, Spring, 2015
This is a tentative outline and will be updated at least 24 hours in advance before each class. Lecture notes are posted under the sections.Date
Jan 14
Chapter 1 – Modeling Change1.1 Modeling Change with Difference Equation Notes
Homework 1: – due Wednesday, Jan 21Page 8: 1(c)(d), 2(c)(d), 3(c)(d), 4(c), 9
Jan 19
MLK Day – No classes
Jan 21
1.2 Approximating Changes with Difference Equation Introduction to MATLABMatLab program: Ex2_1_2.mMatLab program: data1.mMatLab program: data2.m  Notes
Homework 2: – due Wednesday, Jan 28Page 16: 1, 2, 3, 8 
Jan 26
1.3 Solutions to Dynamical SystensNotes Ex1_3_2.mEx1_3_ContvsDis.m 
Homework 3: – due Wednesday, Feb 4Page 31: 1(b)(c)(d), 2(b)(e)(j), 3(a), 4(d),8, 11 
Jan 28
1.4 Systems of Difference EquationsNotes Ex1_4_1.mEx1_4_3.m 
Homework 4:  – due Wednesday, Feb 11Page 52: 4, 6 Project 1.3: – due Wednesday, Feb 11Page 35: 12(1) let a_0=4000, k=0.001, m=100, and M=4000, M=5000, M=6000(2) let k=0.001, m=100, M=5000, and a_0=4000, a_0=5000, a_0=6000Plot results {a_{n+1}} in (1) and (2) in figure(1) and figure(2), respectively.  (3) (extra points) try your choices of m, M, a_0 and k, and give your observations and explanations.
Feb 2
2.1 Mathematical Models 
Feb 4
2.2 Modeling Using Proportionality 
Feb 9
2.3 Modeling Using Geometric Similarity 
Feb 11
3.1 Fitting Models to Data Graphically3.2 Analytic Methods of Model Fitting3.3 Applying the Least-Squares Criterion Notes Ex3_2_Linear.mEx3_2_transform.m
 Homework 5: – due Feb 18 Page 113:3: ln(y)=ln(a)+bln(x)=B+mln(x),(1) plot {(ln(x),ln(y))}(2) use the Least-Squares Method to estimate m and B and plot the line Y=m ln(x)+B in the same plots of (1)(3) compute a and b and plot {(x,y)} and y=ax^b 5(b): ln(P)=ln(a)+bt=B+mt,(1) plot {(t, ln(P))}(2) use the Least-Squares Method to estimate m and B and plot Y=mt+B in the same plots of (1)(3) compute a and b and plot {(t,P)} and P=ae^(bt) Page 121: 2(a), 3, 4(1) use the Simplex Method (Scientific Notebook) to minimize the largest deviation between the data and the model y or P(2) also use the Least-Squares Method to minimize the sum of error squares(3) plot the given data and approximations from (1) and (2)
Feb 16
3.2 Analytic Methods of Model Fitting 
Feb 18
3.3 Applying the Least-Squares Criterion Test 1 – Due Feb 23 solutions
Feb 23
4.1 Harvesting in the Chesapeake Bay and Other One-Term ModelsNotes Ex4_1.mEx4_2.m
Homework 6: – due Wednesday, Feb 25Page 144: 6 using models  (i) y=mx+b  (ii) y=Csqrr(x)  (ii) y=Clog(x+1), extra points for a new modelFor each model:(1) for each model, approximate the model using the least-squares method(2) compute the least-squares error.Which model is better? 
Feb 25
4.2 High Order Polynomials 
Homework 7: – due Wednesday, March 4Page 144: 6 using models  (i) P1   (ii) P2  (iii) P3For each model:(1) approximate the model using the least-squares method; and(2) compute the least-squares error.Which model is better? Page 180: Use Piecewise linear spline modelfor the following problems1(a),  2 Solutions
Mar 2
 44.4 Linear Spline ModelNotes on Linear and Quadratics 
Mar 4
4.4 Quadratic and Cubic Spline ModelNotes on Linear and Quadratics cbspfun.mcbspplot.mcbspcl.mEx4_4_c1.m  
Homework 8: – due Wednesday, March 11Page 180:(1) Use piecewise quadratic spline modelfor 1(a), add a condition y’(7)=2(2) Use cubic spline model (natural spline) for 2. Solutions
Mar 9
 4.4 Cubic Spline Model 
Mar 11
 6.3 Linear Regression Notes
Homework 9: – Wednesday, March 18Page 240: Table 2.7 on Page 1001. predict weight as a linear function of height (w=ah+b) and as a function of a square function of height (w=ah^2+b)2. predict weight as a function of the cube of the height Project 2: Page 184 – Project 7(1) Form the natural spline(2) Sketch the spline and points(3) Use the spline to approximate W(13) and predict W(18)(4) Use the spline to approximate W’(16) and predict W’(18)(5) Approximate integral from 12.5 to 13 of W(h) using the spline Solutions
Mar 16
 11.1 Populating GrowthNotes Ex11_1.m 
Mar 18
11.1 Populating Growth Ex11_2.m 
Homework 10: Wednesday, April 1Page 468:1. a. (let t=t* and P(t)=1/2 M, b. (let t_0=t*, P_0=1/2 M) ,c. (use the equation ln(P/(M-P)=rmt+C and ln(P(t*)/(M-P(t*))=rmt*+C)4.7. (c) Solutions
Mar 23-27
Spring Break  – no classes
Mar 30
11.2 Prescribing Drug Dosage Notes Ex11_3.m 
Apr 1
11.2 Prescribing Drug Dosage 
Homework 11: Wednesday, April 8Page 478: show your work2, 3, 4(also sketch C(t) for [0,3T] using MatLab)7(sketch C(t) for [0,3T] on paper with labeling C_n and R_n)8 (Show your derivations) Solutions
Apr 6
11.4 Graphical Solutions of Autonomous Differential Equations Notes 
Homework 12: Wednesday, April 15Page 490: show your work3, 5, 8, 9Apply the phase line techniques introduced in the text to obtained solution curves for the logistic growth equation:dP/dt=r(M-P)(P-m). Solutions
Apr 8
11.4 Graphical Solutions of Autonomous Differential Equations Test 2 – Due Apr 13
Example 1 – pic1 pic2 Exam 2 Solutions
Apr 13
11.5 Numerical Approximation Methods Notes eulfun.m Ex11_5_1.m  
Apr 15
12.5 Euler’s Method for Systems of Differential Equations Notes Ex12_5_1.m 
Homework 12: Wednesday, April 15Page 497: show your work(1) For each of 1 and 3, compute by hand(i) y0, y1, y2 by Picard’s Method and by Taylor Method, respectively; and(ii) y0, y1, y2 by Euler Method with h=0.1. (2) For each of 1 and 3, approximate y(t) by Euler method as follows:1, 1<=x<=5, h=0.4, 0.2, 0.1 and 0.05; and3, 0<=x<=2, h=0.4, 0.2, 0.1 and 0.05.For each one, sketch the sequences in one figure and label each curve. Page 565: 4(1) Compute Y_1 by hand and the exact approximation error.(2) Compute Y(t) for 0<=t<=2, and plot (t,x(t)) and (t,y(t)) in one figure and (x(t),y(t)) in another figure. Solutions 
Apr 20
12.1 Graphical Solutions of Autonomous Systems of 1st-OrderDifferential Equations  
Apr 22
12.2 A Competitive Hunter Model 
Homework 13: Monday, April 15 Page 529:(1) 2, 6, 7,(2) Compute the exact solution of 6.(3) 9(a)(c) Solutions 
Apr 27
12.3 A Predator-Prey Model 
Apr 28
12.4 Two Military Examples 
May 4Mon
 Final Exam – 1:00pm – 4:00pm
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