# MATHS 361 : Partial Differential Equations

## Science

### Course Prescription

Partial differential equations (PDEs) are used to model many important applications of phenomena in the real world such as electric fields, diffusion and wave propagation. Covers: linear PDEs and analytical methods for their solution, weak solutions. Recommended preparation: MATHS 253.

### Course Overview

This is an introductory course in partial differential equations (PDEs). We cover Fourier series, Fourier integrals, boundary value problems, separation of variables, Laplace and Fourier transform solutions, distributions, and Green’s functions, with application to the solution of second order PD’s in one, two and three dimensions. The course is appropriate for third year students wanting a thorough grounding in analytic methods for PDEs, including engineering and physics students, as well as preparation for postgraduate study in applied mathematics.

### 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

### Learning Outcomes

By the end of this course, students will be able to:
1. Set up suitable boundary conditions for the Laplace, diffusion and wave equations in 2 and 3 dimensions. (Capability 1, 2 and 3)
2. Solve the Laplace, difussion and wave equations in 2 and 3 dimensions analytically for simply posed Neuman and Dirichelet problems with possibly time varying inhomogeneous boundary conditions. (Capability 1, 2 and 3)
3. Demonstrate a good understanding of the use of Fourier series and transform function representations to solve PDEs. (Capability 1, 2 and 3)
4. Solve simple PDEs using eigenfunction expansions and eigenfunction transforms. (Capability 1, 2 and 3)
5. Find and critically evaluate weak solutions for partial differential equations. (Capability 1 and 2)
6. Use distributions on simple applications. (Capability 1, 2 and 3)
7. Demonstrate a good understanding of the characteristics of the solutions to the Laplace, diffusion and wave equation in 2 and 3 dimensions. (Capability 1 and 2)

### Assessments

Assessment Type Percentage Classification
Final Exam 50% Individual Examination
Test 20% Individual Test
Assignments 30% Individual Coursework
1 2 3 4 5 6 7
Final Exam
Test
Assignments

### Tuākana

Maori and Pacific students are encouraged to participate in the Maths Tuakana Programme. For information please visit the website
www.math.auckland.ac.nz/en/for/maori-and-pacific-students.html

### Key Topics

1. Introduction, PDE's and boundary conditions. Modelling the diffusion (heat) equation. Introduction to separation of variables.
2. Fourier Series and separation of variables. Orthogonality of functions and sets of functions. Real trig series. Convergence and sketching Fourier series. Complex Fourier series.
3. Sturm-Liouville Problems. Eigenvalues and eigenfunctions.  Sturm-Liouville eigenvalue problems, existence and orthogonality of solutions, eigenfunction expansions.
4. Separation of variables. Separation of variables for the wave equation and Laplace's equation, in several geometries.
5. Wave Equation.  D'Alembert's solution.
6. Laplace Transforms. Introduction to transform methods. Calculation and properties of the transform. Solution of ODEs and PDEs.
7. Fourier Transforms. Fourier representation of delta function. Convolution theorem. Application to PDEs.
8. Distributions. Basics of distributions.
9. Green's functions. Deriviation of Green's representation, application to PDEs.

### Special Requirements

No special requirements

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.

### Delivery Mode

#### Campus Experience

Lectures will be available as recordings.
The exam and mid-semester test will be held on campus.
The activities for the course are scheduled as a standard weekly timetable.

### Learning Resources

The text is ‘Mathematical Methods for Engineers and Scientists, Volume 3’, by K. T. Tang. We recommend you read the textbook. The book is available electronically on the library website. The book contains more information than will be covered in the course.

### 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.

### 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 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.

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 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.

Level 1:  Delivered normally as specified in delivery mode
Level 2: You will not be required to attend in person.  All teaching and assessment will have a remote option.
Level 3 / 4: All teaching activities and assessments are delivered remotely.

### 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 .

### 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 26/01/2021 09:32 a.m.