MATH4PD Numerical Solution of PDEs
Partial differential equations (PDEs) are used to model a wide range of phenomena in engineering, economics and the natural sciences. They generalise ordinary differential equations (ODEs) modelling functions of one vari- able to describe quantities depending on multiple variables, such as time and space, or several spatial dimensions. Examples are the heat equation, the Laplace equa- tion and the wave equation. Because closed-form analytical solutions can seldom be found, even for the simplest PDEs, approximate solutions are usually sought. This course gives an overview of approximation methods for solving PDEs on finite domains, the so-called boundary-value problems, ranging from the semi-analytical separation of variable technique to the numerical finite difference method. The focus of the paper will be on implementing methods to generate approximate solutions using computational software Matlab and analyse their properties.
The following topics will covered
- Method of characteristics
- Separation of variables
- Finite difference method
- Green's function method
- Finite element method (if time permits)
2019, Semester 1.
MATH202 and MATH203. COMO204, COMO303 and MATH304 are recommended. Some experience using the computational software Matlab would be helpful.
Fabien Montel (firstname.lastname@example.org).
The required text for this course is Introduction to partial differential equations by P. Olver. You can download a free copy of the book at this link.
The internal assessment consists of 3 or 4 long assignments, as part of which you will be asked to solve numerically a number of initial/boundary-value problems and write a report investigating the behaviour of solutions for different values of the parameters. We encourage you to start early.
Your final mark F in the paper will be calculated according to this formula:
F = A
- A is the Assignments mark
and all quantities are expressed as percentages.
Students must abide by the University’s Academic Integrity Policy
Academic integrity means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.
Academic misconduct is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.
Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.
All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.
Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.
If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.
Types of academic misconduct are as follows:
The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).
- Although not intended, unintentional plagiarism is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided.
- Intentional plagiarism is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.
Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..
Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.
Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.
Use of Unauthorised Materials
Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.
Assisting Others to Commit Academic Misconduct
This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.