MATHS 362 : Methods in Applied Mathematics


2020 Semester Two (1205) (15 POINTS)

Course Prescription

Covers a selection of techniques including the calculus of variations, asymptotic methods and models based on conservation laws. These methods are fundamental in the analysis of traffic flow, shocks, fluid flow, as well as in control theory, and the course 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; Calculus of Variations; Asymptotic Analysis.
The course may be used as a part of a mathematics major, as well as, towards the pathway in applied mathematics and it would be a useful addition to a pure mathematics or physics pathway and 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 1: Disciplinary Knowledge and Practice
Capability 2: Critical Thinking
Capability 3: Solution Seeking
Capability 5: Independence and Integrity
Graduate Profile: Bachelor of Science

Learning Outcomes

By the end of this course, students will be able to:
  1. Apply the basic theory of calculus of variations to solve optimisation problems. (Capability 1, 2, 3 and 5)
  2. Apply the method of characteristics to solve first order partial differential equations (Capability 1, 2, 3 and 5)
  3. Apply the method of characteristics to traffic flow problems (Capability 1, 2, 3 and 5)
  4. Demonstrate the use of conservation laws to derive a model of traffic flow (Capability 1, 2, 3 and 5)
  5. Apply asymptotic analysis to derive solutions of differential equations depending on a small parameter (Capability 1, 2, 3 and 5)
  6. Use the concept of boundary layers to solve differential equations (Capability 1, 2, 3 and 5)


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


Maori and Pacific students are encouraged to participate in the Maths Tuakana Programme. For information please visit the website

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.
  • Calculus of Variations. This is about finding the minimum or maximum value of an integral which involves an unknown function and its derivatives. The problem is to find the unknown function that minimises or maximises the integral. We'll see that the unknown function can be found by solving an ordinary differential equation. 
  • 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.

Learning Resources

Texts. The texts for this course are ebooks. You can get them from the university library's website.

  • Mathematical Methods for Engineers and Scientists, Volume 3, by K. T. Tang. We use the last chapter, Calculus of Variations.
  • 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.

Special Requirements

No special requirements

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 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.

Digital 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.


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.

Academic Integrity

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework 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. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

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 at

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:

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

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.

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 (


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.

Published on 09/07/2020 03:23 p.m.