## APP MTH 3021 - Modelling with Ordinary Differential Equations III

### North Terrace Campus - Semester 1 - 2020

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, linearisation 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.

• General Course Information
##### Course Details
Course Code APP MTH 3021 Modelling with Ordinary Differential Equations III School of Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Up to 3 contact hours per week. Y MATHS 2102 or MATHS 2106 or MATHS 2201 MATHS 2104 or MATHS 2107 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, linearisation 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 Staff

Course Coordinator: Dr Edward Green

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
Students who successfully complete the course will:
1. understand how to model dynamical (time-varying) systems using ordinary differential equations;
2. be able to identify and analyse stability of equilibrium solutions;
3. be able to solve ordinary differential equations numerically;
4. be able to analyse the effect of parameters on the structure of solutions;
5. understand analytical solution theory for linear systems of ordinary differential equations;
6. appreciate the necessity of numerical and qualitative methods for analysing solutions for nonlinear systems;
7. have a detailed understanding of several ordinary differential equations models arising in physics, engineering, biology and other applications;
8. be able to apply the calculus of variations to find optimal solutions to problems;
9. appreciate the derivation of many physical laws from variational principles.

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
1,2,4,6,8,9
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
1,4,6,9
• 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
all
• Learning Resources
None.
##### Recommended Resources
1. 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 Learning
This 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.

• Learning & Teaching Activities
##### Learning & Teaching Modes
This 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 Summary
Lectures include
1. Modelling examples, different types of questions (and how to answer them), necessity for theory and computation.
2. Systems of first-order equations, existence and uniqueness of solutions.
3. Numerical error, well and ill-conditioned problems.
4. Stable and unstable algorithms, conditioning and stability.
5. Numerical solution of 1st order initial value problems (IVPs), finite difference approximation.
6. Explicit and implicit methods, forward and backward Euler methods, consistency, convergence.
7. Stability of forward and backward Euler methods, the stability region of a numerical method.
8. Optimal step size, stiff problems.
9. Matlab ODE solvers and non-linear IVPs.
10. Predictor-corrector and Runge-Kutta schemes.
11. Systems of IVPs, second-order IVPs.
12. Numerical solution of boundary value problems.
13. Qualitative analysis. Revision of autonomous scalar equations, fixed points, phase line analysis, stability criteria.
14. Fishing models, examples of bifurcations.
15. Autonomous systems in two dimensions. Analysis of linear systems in two dimensions, the phase plane.
16. Nonlinear systems and linearisation, the Hartman-Grobman theorem, general population interaction model, Lotka-Volterra predator-prey equations.
17. Mutualism and competition population models, analysis of the Kermack-McKendrick epidemic model.
18. Limit cycles, bifurcations in 2D systems, Hopf bifurcations.
19. Examples of applications.
20. Introduction to the calculus of variations: motivations, some important problems.
21. Preliminary material: revision of extrema of multivariate functions, multivariate Taylor series, Lagrange multipliers
22. Formulation of variational problems, functionals. Euler-Lagrange equation. Special case: F = F(y').
23. Special case F = F(y, y') (autonomous). Brachistrochrone.
24. Further examples: Fermat's principle, Principle of Least Action
25. Several independent variables: surface area minimisation
26. Integral constraints: isoperimetric problems, Dido's problem, catenary of fixed length
27. Isoperimetric problems continued: Dido's problems with parameterised solution, catenary of fixed length

Tutorials include

1. Ill- and well-conditioned problems, stable and unstable algorithms, numerical error.
2. Stiff problems, Matlab ODE solvers, regions of stability for explicit and implicit Euler methods, numerical solution of vector IVPs.
3. One-dimensional models: equilibria and their stability, phase-line analysis, bifurcation diagrams, classification of bifurcations.
4. Two-dimensonal models, linearisation of non-linear models, phase portraits.
5. Development, analysis and intrepretation of models
6. Applications of the Euler-Lagrange equations
##### Specific Course Requirements
Understanding 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 Experience
A group project with a written report develops research skills, teamwork skills, and communication skills.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Component Weighting Objective Assessed Assignments 20% all Project 10% 1,2,3,4 Exam 70% all
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Due to the current COVID-19 situation modified arrangements have been made to assessments to facilitate remote learning and teaching. Assessment details provided here reflect recent updates.

(Details taken from "4010_APP_MTH_X_0003")

The new assessment scheme for this course is:

Assignments and individual project 50%
Online exam in exam period 50%
The breakdown is as follows:

Assignment 1 will continue be weighted as 5% of the grade for the course (as previously advised). Due date: Week 4 - 27 March.

Assignments 2, 3 & 4 will be weighted higher, to comprise 25% of the grade for the course in total (i.e. each assignment is worth 8.33%). Due dates: Assignment 2 - Week 6; Assignment 3 - Week 8; Assignment 4 - Week 12.

The individual project will be weighted higher, at 20% of the course grade. Due date: Week 10.

The examination will be replaced with an online exam / test which will be scheduled during the usual exam period. This online test will be worth 50% of the marks for the course. I will provide further details about the nature of the examination later.
##### Assessment Related Requirements
An 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 Detail
 Assessment Item Weighting Assignment 1 5% Assignment 2 5% Assignment 3 5% Assignment 4 5% Project 10%
##### Submission
1. All assignments are to be submitted online via MyUni.
2. 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)
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.

• Student Feedback

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.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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