# MATHS 762 : Nonlinear Partial Differential Equations

## Science

### Course Prescription

A study of exact and numerical methods for non-linear partial differential equations. The focus will be on the kinds of phenomena which only occur for non-linear partial differential equations, such as blow up, shock waves, solitons and special travelling wave solutions.

### Course Overview

A study of exact and numerical methods for non-linear partial differential equations. The focus will be on the kinds of phenomena which only occur for non-linear partial differential equations, such as blow up, expansion waves, shock waves, solitons, and special travelling wave solutions. This is a core offering for honours in applied mathematics and may be of interest to students majoring in Maths, Physics, or Computer Science. Required preparation: B- in both MATHS 340 and 361, or equivalent.

### Course Requirements

Prerequisite: B- in both MATHS 340 and 361

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

### Learning Outcomes

By the end of this course, students will be able to:
1. Derive, analyse and solve simple nonlinear equations with characteristics (Capability 1, 2 and 3)
2. Derive, analyse and solve nonlinear equations with two characteristics. (Capability 1, 2 and 3)
3. Understand Reaction-Diffusion equations (Capability 1, 2 and 3)
4. Produce Solitons and numerical and analytical solutions to the Korteweg-de Vries Equation (Capability 1, 2 and 3)
5. Demonstrate an understanding of different solution types for integrable systems, including the KdV and NLS, and the concepts surrounding solitons. (Capability 1, 2 and 3)
6. Write simple MATLAB code to solve certain nonlinear pde’s (Capability 1, 2 and 3)
7. Be able to present complete and coherent assignment answers. (Capability 1, 2, 4 and 5)
8. Derive, analyse and solve the nonlinear water wave equations (Capability 1, 2 and 3)
9. Analyse the nonlinear expansion waves in real-world problems (Capability 1, 2 and 3)
10. Analyse the shock waves in the real-world problems (Capability 1, 2 and 3)

### Assessments

Assessment Type Percentage Classification
Assignments 40% Individual Coursework
Final Exam 60% Individual Examination
1 2 3 4 5 6 7 8 9 10
Assignments
Final Exam

### Learning Resources

Texts recommended
• Wave Motion by Billingham and King.
• Solitons and the Inverse Scattering Transform by Ablowitz and Segur
• Solitons, Nonlinear Evolution Equations and Inverse Scattering by Ablowtiz and Clarkson
• Spectral methods in MATLAB by Trefethen
Webpage Many useful resources are on the wiki: http://www.wikiwaves.org/Category:Nonlinear_PDE’s_Course

### Special Requirements

None.

It is expected that you spend 10 hours per week working on this course. The normal pattern of student study is expected to be (on average, each week):

2 hours lectures;

2 hour lab, including preparation;

3 hours lecture preparation and review;

3 hours assignments and exam preparation.

Students are expected to participate in class activites. After each class you should review the material from the class and try any recommended examples. You are expected to preview the material provided for each lecture before you come to class.

### 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.

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 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.

### 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 (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.

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