APP MTH 7035 - Modelling with Ordinary Differential Equations
North Terrace Campus - Semester 1 - 2020
General Course Information
Course Code APP MTH 7035 Course Modelling with Ordinary Differential Equations Coordinating Unit School of Mathematical Sciences Term Semester 1 Level Postgraduate Coursework Location/s North Terrace Campus Units 3 Contact Up to 3 hours per week Available for Study Abroad and Exchange Y Assumed Knowledge (MATHS 2102 or MATHS 2106 or MATHS 2201) and (MATHS 2104 or MATHS 2107) Course Description Differential equation models describe a wide range of complex problems in biology, engineering, physical sciences, economics and finance. This course focuses on ordinary differential equations (ODEs) and develops students' skills in the formulation, solution, understanding and interpretation of coupled ODE models. A range of important biological problems, from areas such as resource management, population dynamics, and public health, drives the study of analytical and numerical techniques for systems of nonlinear ODEs. A key aim of the course is building practical skills that can be applied in a wide range of scientific, business and research settings.
Topics covered are: analytical methods for systems of ODEs, including vector fields, fixed points, phase-plane analysis, linearization of nonlinear systems, bifurcations; general theory on existence and approximation of ODE solutions; biological modelling; explicit and implicit numerical methods for ODE initial value problems, computational error, consistency, convergence, stability of a numerical method, ill-conditioned and stiff problems.
Course Coordinator: Dr Edward Green
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning OutcomesStudents who successfully complete the course will:
- understand how to model dynamical (time-varying) systems using ordinary differential equations;
- be able to identify and analyse stability of equilibrium solutions;
- be able to solve ordinary differential equations numerically;
- be able to analyse the effect of parameters on the structure of solutions;
- understand analytical solution theory for linear systems of ordinary differential equations;
- appreciate the necessity of numerical and qualitative methods for analysing solutions for nonlinear systems;
- have a detailed understanding of several ordinary differential equations models arising in physics, engineering, biology and other applications;
- be able to apply the calculus of variations to find optimal solutions to problems;
- appreciate the derivation of many physical laws from variational principles.
University Graduate Attributes
This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:
University Graduate Attribute Course Learning Outcome(s) Deep discipline knowledge
- informed and infused by cutting edge research, scaffolded throughout their program of studies
- acquired from personal interaction with research active educators, from year 1
- accredited or validated against national or international standards (for relevant programs)
all Critical thinking and problem solving
- steeped in research methods and rigor
- based on empirical evidence and the scientific approach to knowledge development
- demonstrated through appropriate and relevant assessment
1,2,4,6 Teamwork and communication skills
- developed from, with, and via the SGDE
- honed through assessment and practice throughout the program of studies
- encouraged and valued in all aspects of learning
1,4,6 Career and leadership readiness
- technology savvy
- professional and, where relevant, fully accredited
- forward thinking and well informed
- tested and validated by work based experiences
all Self-awareness and emotional intelligence
- a capacity for self-reflection and a willingness to engage in self-appraisal
- open to objective and constructive feedback from supervisors and peers
- able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
Recommended Resources1. Strogatz, Steven. Nonlinear Dynamics and Chaos (Perseus, 2001)
2. Trefethen, Lloyd N., Birkisson, Ásgeir and Driscoll, Tobin A. Exploring ODEs (SIAM, 2018)
3. Riley, Ken F., Hobson, Michael P. and Bence, Stephen J. Mathematical Methods for Physics and Engineering (Cambridge, 2006)
4. Weinstock, Robert. Calculus of Variations with applications to physics and engineering (Dover, 1974)
5. Edelstein-Keshet, Leah. Mathematical Models in Biology (SIAM, 2005)
6. Butcher, John. Numerical Methods for Ordinary Differential Equations (Wiley, 2008)
7. Chicone, Carmen. Ordinary Differential Equations with Applications (Springer, 2006)
8. Dahlquist, Germund and Bjorck, Ake. Numerical Methods (Dover, 2003)
Online LearningThis course uses MyUni (Canvas) exclusively for providing electronic resources, such as lecture notes, assignment papers, sample solutions, discussion boards, etc. It is recommended that the students make appropriate use of these resources.
Link to MyUni login page: https://myuni.adelaide.edu.au
Learning & Teaching Activities
Learning & Teaching ModesThis course relies on lectures as the primary delivery mechanism for the material. Tutorials supplement the lectures by providing exercises and example problems to enhance the understanding obtained through lectures. A sequence of written assignments provides assessment opportunities for students to gauge their progress and understanding. Some computer programming in Matlab will also be required to promote understanding of computational methods.
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload Hours Lectures 30 90 Tutorials 6 18 Assignments 4 28 Project 1 20 Total 156
Learning Activities SummaryLecture Outline
- Modelling examples, different types of questions (and how to answer them), necessity for theory and computation.
- Systems of first-order equations, existence and uniqueness of solutions.
- Numerical error, well and ill-conditioned problems.
- Stable and unstable algorithms, conditioning and stability.
- Numerical solution of 1st order initial value problems (IVPs), finite difference approximation.
- Explicit and implicit methods, forward and backward Euler methods, consistency, convergence.
- Stability of forward and backward Euler methods, the stability region of a numerical method.
- Optimal step size, stiff problems.
- Matlab ODE solvers and non-linear IVPs.
- Predictor-corrector and Runge-Kutta schemes.
- Systems of IVPs, second-order IVPs.
- Numerical solution of boundary value problems.
- Qualitative analysis. Revision of autonomous scalar equations, fixed points, phase line analysis, stability criteria.
- Fishing models, examples of bifurcations.
- Autonomous systems in two dimensions. Analysis of linear systems in two dimensions, the phase plane.
- Nonlinear systems and linearisation, the Hartman-Grobman theorem, general population interaction model, Lotka-Volterra predator-prey equations.
- Mutualism and competition population models, analysis of the Kermack-McKendrick epidemic model.
- Limit cycles, bifurcations in 2D systems, Hopf bifurcations.
- Examples of applications.
- Introduction to the calculus of variations: motivations, some important problems.
- Preliminary material: revision of extrema of multivariate functions, multivariate Taylor series, Lagrange multipliers
- Formulation of variational problems, functionals. Euler-Lagrange equation. Special case: F = F(y').
- Special case F = F(y, y') (autonomous). Brachistrochrone.
- Further examples: Fermat's principle, Principle of Least Action
- Several independent variables: surface area minimisation
- Integral constraints: isoperimetric problems, Dido's problem, catenary of fixed length
- Isoperimetric problems continued: Dido's problems with parameterised solution, catenary of fixed length
- OIll- and well-conditioned problems, stable and unstable algorithms, numerical error.
- Stiff problems, Matlab ODE solvers, regions of stability for explicit and implicit Euler methods, numerical solution of vector IVPs.
- One-dimensional models: equilibria and their stability, phase-line analysis, bifurcation diagrams, classification of bifurcations.
- Two-dimensonal models, linearisation of non-linear models, phase portraits.
- Development, analysis and intrepretation of models.
- Applications of the Euler-Lagrange equations.
Specific Course RequirementsUnderstanding of and ability to use analytic solution methods for first-order and second-order differential equations as taught in Differential Equations II or Engineering Mathematics IIA.
Ability to write a simple Matlab code from scratch, for example, to solve a first-order initial value problem using Euler's method. Knowledge of numerical methods to the level taught in Numerical Methods II is assumed.
Small Group Discovery ExperienceA group project with a written report develops research skills, teamwork skills, and communication skills.
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Component Weighting Objective Assessed Assignments 20% all Project 10% 1,2,3,4 Exam 70% all
Assessment Related RequirementsAn aggregate score of at least 50% is required to pass the course.
Some computer programs will need to be written that will form part of the assignment assessment.
Assessment Item Weighting Assignment 1 5% Assignment 2 5% Assignment 3 5% Assignment 4 5% Project 10%
- All assignments are to be submitted online via MyUni.
- Assignments will have a two week turn-around time for feedback to students.
Grades for your performance in this course will be awarded in accordance with the following scheme:
M10 (Coursework Mark Scheme) Grade Mark Description FNS Fail No Submission F 1-49 Fail P 50-64 Pass C 65-74 Credit D 75-84 Distinction HD 85-100 High Distinction CN Continuing NFE No Formal Examination RP Result Pending
Further details of the grades/results can be obtained from Examinations.
Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.
Final results for this course will be made available through Access Adelaide.
The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.
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