## APP MTH 7035 - Modelling with Ordinary Differential Equations

### North Terrace Campus - Semester 1 - 2024

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.

• General Course Information
##### Course Details
Course Code APP MTH 7035 Modelling with Ordinary Differential Equations Mathematical Sciences Semester 1 Postgraduate Coursework North Terrace Campus 3 Up to 3 hours per week Y (MATHS 2102 or MATHS 2106 or MATHS 2201) and (MATHS 2104 or MATHS 2107) Ongoing assessment, exam
##### 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)

Attribute 1: Deep discipline knowledge and intellectual breadth

Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

all

Attribute 2: Creative and critical thinking, and problem solving

Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

1,2,4,6

Attribute 3: Teamwork and communication skills

Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.

1,4,6

Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.

all

Attribute 8: Self-awareness and emotional intelligence

Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.

all
• Learning Resources
None.
##### Recommended Resources
1. Strogatz, Steven. Nonlinear Dynamics and Chaos (Chapman&Hall, 2015)
2. Weinstock, Robert. Calculus of Variations with applications to physics and engineering (Dover, 1974)
3. Edelstein-Keshet, Leah. Mathematical Models in Biology (SIAM, 2005)
4. Butcher, John. Numerical Methods for Ordinary Differential Equations (Wiley, 2008)
##### 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 topic videos as the primary delivery mechanism for the material. Tutorials and workshops are the primary direct contact hours, during which students will both reinforce and employ the understanding obtained through lectures. Weekly quizzes provide regular 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 Topic videos & quizzes ~25 hours 86 Tutorials 10 20 Workshops 12 12 Assignments 3 18 Project 1 20 Total 156
##### Learning Activities Summary
Introduction
• Modelling examples, different types of questions (and how to answer them), necessity for theory and computation.
Numerical solution of ordinary differential equations
• Sources of error in solutions
• Well- and ill-posed problems
• Well and ill-conditioned problems
• Finite difference schemes
• Consistency, stability and convergence
• Explicit and implicit methods
• Optimal step size, stiff problems.
• Matlab ODE solvers and non-linear IVPs.
Developing and analysing ordinary differential equation models
• Autonomous systems in two dimensions. Analysis of linear systems in two dimensions, the phase plane.
• Nonlinear systems and linearisation, the Hartman-Grobman theorem
• Limit cycles, bifurcations in 2D systems, Hopf bifurcations
• Applications - including modelling interacting populations, models for epidemics

The calculus of variations
• Motivation: finding the shortest distance between two points, the shape of a hanging chain and other important problems.
• Functionals
• Formulation of variational problems
• Derivation of the Euler-Lagrange equation
• Special-case solutions of the Euler-Lagrange equation: geodesics, Fermat's principle, Principle of Least Action
• Generalising the Euler-Lagrange equation to the case of several dependent variables
• Problems with integral constraints: isoperimetric problems, Dido's problem, catenary of fixed length
• Generalisating the Euler-Lagrange equation to the case of several independent variables
Tutorials and workshops will provide students with opportunities to work together to broaden their knowledge and practice applying the course material.

##### 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.
• 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 Quizzes 5% all Assignments 15% all Project 15% 1-7 Mid-semester test 15% 1-7 Exam 50% all

##### 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 assessment for the project and some assignments.
##### Assessment Detail
 Assessment Item Released Due Weighting Assignment 1 week 3 week 5 5% Assignment 2 week 5 week 7 5% Assignment 3 week 10 week 13 5% Project week 7 week 11 15%
##### 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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