MATHS 362 : Methods in Applied Mathematics


2023 Semester Two (1235) (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 and Asymptotic Analysis.

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 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
A minimum of 35% in the exam is required to pass the course.


Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at

Māori and Pacific students are encouraged to participate in the Maths Tuākana 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.

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.

  • 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.
The texts for this course are ebooks, which you can get them from the university library's website.

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.


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.

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.


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

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.

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 course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, 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


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 students may be asked to submit 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. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 28/10/2022 11:28 a.m.