Methods of Applied Mathematics: Honors Mathematics 450 and 451 Each 3 Credit Hours

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GENERAL DESCRIPTION

In this course, students perform and analyze physical experiments in the context of an advanced mathematics course. This capstone course integrates the students' experience with mathematical modeling, mathematical analysis, numerical methods, computation, engineering and communication. In the first semester, students have short modules (2-4 weeks) that include relatively simple experiments and numerical simulations. This prepares students for the second semester, when students work in teams to perform and analyze experiments of greater complexity using more advanced mathematical skills. At the end of the second semester, students present their research results both orally and in writing.

FALL SEMESTER

Texts

Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics and Traffic Flow

Farlow, Partial Differential Equations for Scientists and Engineers

Experimental Apparatus

Vernier LabPro--Data acquisition and analysis software, Accelerometer, Photogates, Temperature probe, Masses, Springs, Pendulum, Cycloid track, Power supply, voltmeter, conductive paper and pens

Syllabus

Unit I: Introduction--Math Modeling, Gravity and Newton's Law of Cooling

Week 1: Review of Differential Equations, Introduction to Mathematical Modeling and Applied Problems

Physical Experiment 1: Newton's Law of Cooling--is the power really 1?

Week 2: Equilibrium and Stability in one dimension (1st order), Newton's Law of Cooling Review vector calculus, Newton's laws, conservative systems

Week 3: Least squares fitting for realistic data

Project 1: Mathematical modeling and Newton's Law of Cooling experiment analysis

Unit II: Mechanics I--The Brachistochrone

Week 4: Calculus of Variations

Week 5: Derivation of the Nonlinear Differential Equation governing the Brachistochrone (Curve for which a ball travels from one point to another in the fastest time under the influence only of gravity), Solution to the Nonlinear Ordinary Differential Equation (Parametric Equations)

Physical Experiment 2: Timing a trajectory: the Brachistochrone vs. the line

Week 6: Tautochrone property of the Solution, Analysis for the line and of the cycloid for different height/length ratios

Project 2: Calculus of variations, Brachistochrone experiment and analysis of the cycloid

Week 7: Review and Midterm and Going over Midterm

Unit III: Mechanics II--Mass-Spring Systems

Week 8: Review Midterm, Second order ODEs and harmonic motion, Dimensional Analysis

Week 9: Derivation and solution of undamped and damped single mass-spring systems

Physical Experiment 3: Single vertical mass-spring setup

Week 10: Phase plane analysis, Double mass-spring system, Non-linear oscillations and the Pendulum

Project 3: Measuring the spring constant, frequency and evaluating linearity of a spring and other mass-spring analysis

Week 11: Linear Stability and Linearization (higher order), Energy Conservation and Energy Curves, Numerical Methods for ODEs Physical Experiment 4: Double mass-spring and its frequencies Project 4: Double mass-spring and its frequencies; how initial conditions influence the dynamics of the double mass-spring; nonlinear springs

Week 12: Phase curves for the damped pendulum, The Spring Pendulum Project + Physical Experiment 5: Timing the pendulum, analysis of the nonlinear pendulum and linearized pendulum equations

Unit IV: Electrostatics and Incompressible Fluids

Week 13: Derivation of Laplace equation for potential flow, Electrostatic potential, Properties of the Laplace equation, Elliptic PDEs

Week 14: Separation of Variables, Solutions in Rectangular and Cylindrically symmetric regions

Week 15: Finite difference methods, Review Physical Experiment 6: Electrostatic Field Mapper experiment Project 6: Analytic and Experimental Solution of Laplace's equation for electrostatics problems (equipotential and flux lines)

Grading Policy

The final grade in this course will be determined as follows:

Homework/Projects: 66%

Midterm and Final Exams: 34%

SPRING SEMESTER

General Description

In the spring semester, students learn more advanced methods from classical mechanics and use them to study problems that have attracted more recent interest: dynamical bias in coin tosses, as shown by Diaconis et al. …