APP MTH 7035 - Modelling with Ordinary Differential Equations III
North Terrace Campus - Semester 1 - 2015
General Course Information
Course Code APP MTH 7035 Course Modelling with Ordinary Differential Equations III Coordinating Unit Applied Mathematics 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 Prerequisites (MATHS 2101 and MATHS 2102) or (MATHS 2201 and MATHS 2202) Incompatible APP MTH 3013, APP MTH 3004 Assumed Knowledge MATHS 2104 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: Professor Yvonne Stokes
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning OutcomesStudents who successfully complete the course should:
- understand how to model time-varying systems using ordinary differential equations
- be able to identify and analyse stability of equilibrium solutions
- be able to numerically solve ordinary differential equations
- be able to analyse how the structure of solutions can change depending on a parameter
- understand the 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, biology and chemistry, namely oscillator models, Lotka-Volterra competition and predator-prey models, Michaelis-Menton kinetics and SIR epidemic models
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) Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. all The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 1 An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 2,3,4,5,6,7 A proficiency in the appropriate use of contemporary technologies. 3,7 A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. all
Recommended Resources1. Butcher, John. Numerical Methods for Ordinary Differential Equations (Wiley, 2008)
2. Chicone, Carmen. Ordinary Differential Equations with Applications (Springer, 2006)
3. Dahlquist, Germund and Bjorck, Ake. Numerical Methods (Dover, 2003)
4. de Vries, Gerda et al. A Course in Mathematical Biology (SIAM, 2006)
5. Edelstein-Keshet, Leah. Mathematical Models in Biology (SIAM, 2005)
6. Strogatz, Steven. Nonlinear Dynamics and Chaos (Perseus, 2001)
Online LearningThis course uses MyUni 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/webapps/login/
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.
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 5 35 Mid-semester test 1 5 Total 148
Learning Activities SummaryLecture Outline
- Modelling examples, necessity for theory and computation.
- Autonomous scalar equations, fixed points, phase line analysis, stability criteria, fisheries management model.
- Nondimensionalisation, constant yield fishing model, bifurcation, saddle-node bifurcation.
- Constant effort fishing model, transcritical bifurcaiton.
- Supercritical pitchfork bifurcation.
- Subcritical pitchfork bifurcation, hysteresis.
- The spruce-budworm model.
- A bifurcation diagram in a two parameter space.
- Exact solution of 1D models, existence and uniqueness.
- Existence and uniqueness theorem.
- 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 methos, forward and backward Euler methods, consistence, convergence.
- Stability of forward and backward Euler methods, the stability region of a numerical method.
- Optimal step size, stiff problems.
- More numerical solution methods, Matlab ODE solvers, non-linear IVPs.
- Predictor-corrector and Runge-Kutta schemes.
- Systems of IVPs, second-order IVPs.
- Numerical solution of boundary value problems
- Linear autonomous systems in two dimensions: some models.
- The Kermack-McKendrick epidemic model, existence and uniqueness of solutions, the phase plane.
- Analysis of linear systems in two dimensions.
- 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 bifurcation.
- Chemical kinetics, Michaelis-Menten kinetics.
- Linear nonautonomous systems in higher dimensions: homogeneous systems.
- Linear nonautonomous systems: nonhomogeneous systems.
- Course summary and revision.
- One-dimensional models: scaling, equilibria and their stability, bifurcation, benefits of numerical and analytic solution.
- Phase-line analysis, bifurcation diagrams, classification of bifurcations.
- Understanding a model, bifurcation analysis and interpretation, hysteresis.
- Ill- 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.
- Two-dimensonal models, linearisation of non-linear models, phase portraits.
Specific Course RequirementsNone.
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 Assigments 20% all Mid-semester test 10% 1,2,3,4 Exam 70% all
Assessment Related RequirementsAn aggregate score of at least 50% is required to pass the course.
Assessment Item Distributed Due Date Weighting Assignment 1 Week 3 Week 4 4% Assignment 2 Week 5 Week 6 4% Assignment 3 Week 7 Week 8 4% Assignment 4 Week 9 Week 10 4% Assignment 5 Week 11 Week 12 4%
- All written assignments are to be submitted to the designated hand-in boxes within the School of Mathematical Sciences with a signed cover sheet attached.
- Late assignments will not be accepted.
- 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.
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