MATHS 361 : Partial Differential Equations
2022 Semester One (1223) (15 POINTS)
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.
Capabilities Developed in this Course
|Capability 1:||Disciplinary Knowledge and Practice|
|Capability 2:||Critical Thinking|
|Capability 3:||Solution Seeking|
- Set up suitable boundary conditions for the Laplace, diffusion and wave equations in 2 and 3 dimensions. (Capability 1, 2 and 3)
- Solve the Laplace, Difusion 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)
- Demonstrate a good understanding of the use of Fourier series and transform function representations to solve PDEs. (Capability 1, 2 and 3)
- Solve simple PDEs using eigenfunction expansions and eigenfunction transforms. (Capability 1, 2 and 3)
- Find and critically evaluate weak solutions for partial differential equations. (Capability 1 and 2)
- Use distributions on simple applications. (Capability 1, 2 and 3)
- 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)
- Demonstrate the use of Green's functions to solve simple types of ordinary differential equations. (Capability 1, 2 and 3)
|Final Exam||50%||Individual Examination|
|Assessment Type||Learning Outcome Addressed|
- Introduction, PDE's and boundary conditions. Modelling the diffusion (heat) equation. Introduction to separation of variables.
- Fourier Series and separation of variables. Orthogonality of functions and sets of functions. Real trig series. Convergence and sketching Fourier series. Complex Fourier series.
- Sturm-Liouville Problems. Eigenvalues and eigenfunctions. Sturm-Liouville eigenvalue problems, existence and orthogonality of solutions, eigenfunction expansions.
- Separation of variables. Separation of variables for the wave equation and Laplace's equation, in several geometries.
- Wave Equation. D'Alembert's solution.
- Laplace Transforms. Introduction to transform methods. Calculation and properties of the transform. Solution of ODEs and PDEs.
- Fourier Transforms. Fourier representation of delta function. Convolution theorem. Application to PDEs.
- Distributions. Basics of distributions.
- Green's functions. Derivation of Green's representation, application to ODEs.
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.
Lectures will be available as recordings.
The test and exam will be held on campus.
The activities for the course are scheduled as a standard weekly timetable.
Students are expected to attend lectures and tutorials.
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 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.
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.
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 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.
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
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.
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 https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.
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.