MATHS 362 : Methods in Applied Mathematics

Science

2025 Semester Two (1255) (15 POINTS)

Course Prescription

Covers a selection of techniques to analyse differential equations including the method of characteristics and asymptotic analysis. These methods are fundamental in the analysis of traffic flows, shocks and fluid flows. Introduces foundational concepts to quantify uncertainty in parameters of differential equations and is recommended for students intending to advance in Applied Mathematics.  Recommended preparation: MATHS 253, 361

Course Overview

This course introduces and develops three topics that are important in applied mathematics: First order partial differential equations (PDEs) with application to traffic flow modelling, Parameter Estimation for mathematical models, and Asymptotic Analysis.

Parameter estimation and asymptotic analysis are key tools in mathematical modelling and both are associated with the parameters that arise in a model. In parameter estimation, we use methods of optimisation to find the best parameters for the model to fit data. Asymptotic analysis makes use of small or large values of parameters to find simpler expressions for mathematical models.

The course may be used as a part of a mathematics major, particularly towards the pathway in applied mathematics. It would be a useful addition to a pure mathematics or physics pathway, as well as preparation for postgraduate study in applied mathematics, pure mathematics and physics. In particular, the course is good preparation for students intending to take the partial differential equations courses, Maths 762 and Maths 763, at postgraduate level. The course is useful for students who contemplate a career that involves mathematical modelling. Mathematical modelling is a key skill in physical sciences and in a diverse range of industries, including agricultural, financial, horticultural, manufacturing and energy.

Course Requirements

Prerequisite: MATHS 250, 260

Capabilities Developed in this Course

Capability 2: Sustainability
Capability 3: Knowledge and Practice
Capability 4: Critical Thinking
Capability 5: Solution Seeking
Capability 6: Communication
Capability 7: Collaboration
Graduate Profile: Bachelor of Science

Learning Outcomes

By the end of this course, students will be able to:
  1. Apply the theories of linear and nonlinear least squares estimation to estimate parameters for mathematical models. (Capability 3, 4, 5, 6 and 7)
  2. Apply the method of characteristics to solve first order partial differential equations. (Capability 3, 4, 5, 6 and 7)
  3. Apply the method of characteristics to traffic flow problems. (Capability 2, 3, 4, 5, 6 and 7)
  4. Demonstrate the use of conservation laws to derive a model of traffic flow. (Capability 2, 3, 4, 5, 6 and 7)
  5. Apply asymptotic analysis to derive solutions of differential equations depending on a small parameter. (Capability 3, 4, 5, 6 and 7)
  6. Use the concept of boundary layers to solve differential equations. (Capability 3, 4, 5, 6 and 7)

Assessments

Assessment Type Percentage Classification
Final Exam 50% Individual Examination
Assignments 30% Individual Coursework
Test 20% Individual Test
Assessment Type Learning Outcome Addressed
1 2 3 4 5 6
Final Exam
Assignments
Test
A minimum of 35% in the exam is required to pass the course.

Key Topics

This course introduces and develops three topics that are important in applied mathematics:
  • First order partial differential equations (PDEs) with application to traffic flow. In this topic we'll meet first order partial differential equations and see that they can be solved by solving some associated ordinary differential equations. We'll apply these ideas to a model of vehicle traffic moving along a road. The model exhibits shock waves and rarefaction waves, which occur in a variety of other physical systems.
  • Parameter Estimation. This is about finding the best possible parameters (usually unknown constants) for a mathematical model. we consider how to find the best parameters by minimising the deviation between a model's prediction and a given set of data. 
  • Asymptotic Analysis. Often it's impossible to find an exact formula for a solution of an algebraic or differential equation. If the equation involves a small parameter, it's often possible to get an approximate solution to the problem. This is called asymptotic analysis.

Special Requirements

None.

Tuākana

Whanaungatanga and manaakitanga are fundamental principles of our Tuākana Mathematics programme which provides support for Māori and Pasifika students who are taking mathematics courses. The Tuākana Maths programme consists of workshops and drop-in times, and provides a space where Māori and Pasifika students are able to work alongside our Tuākana tutors and other Māori and Pasifika students who are studying mathematics.

Workload Expectations

This course is a standard 15-point course and students are expected to spend 10 hours per week involved in each 15-point course that they are enrolled in. For each week of this course, you can expect 3 hours of lectures, a 1-hour tutorial, 4 hours of reading and thinking about the content and 2 hours of work on assignments and/or test preparation.

Delivery Mode

Campus Experience

  • Attendance is expected at scheduled activities, including labs/tutorials, to complete components of the course.
  • Lectures will be available as recordings. Other learning activities, including tutorials/labs, will not be available as recordings.
  • The course will not include live online events.
  • The activities for the course are scheduled as a standard weekly timetable delivery.

Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

Texts:
  • Parameter Estimation and Inverse Problems, by Aster, Borchers and Clifford. In particular, Chapter 2 on linear regression, Chapter 9 on nonlinear regression.
  • Introduction to perturbation methods by Mark H. Holmes. This is all about asymptotic analysis. We use only the first two chapters. Alternatively, the following book can be used:
  • Introduction to the Foundations of Applied Mathematics by Mark H Holmes. This is also an ebook. Chapter 5 covers first order PDEs and traffic flow. Chapter 2 covers asymptotic analysis.
The texts for this course are ebooks, which you can get them from the university library's website.
MATHS 362 also makes use of the numerical computing environment and programming language MATLAB and the course includes teaching you how to code in MATLAB. You can download MATLAB for home use by following the instructions here: https://www.software.auckland.ac.nz/en/matlab.html

Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

There are no significant changes. We will be open to ongoing feedback from the 2025 class.

Academic Integrity

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework, tests and examinations as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. A student's assessed work may be reviewed against electronic source material using computerised detection mechanisms. Upon reasonable request, students may be required to provide an electronic version of their work for computerised review.

Class Representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.

Copyright

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website http://disability.auckland.ac.nz

Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

Learning Continuity

In the event of an unexpected disruption we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and your assessment is fair, and not compromised. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator, and if disruption occurs you should refer to the University Website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.

Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.