## MATH304 Partial Differential Equations

Second Semester |

### Paper details

Differential equations are a fundamental mathematical tool for the study of systems that are either in equilibrium or change over time, and are used in most areas of science, engineering, and mathematics. This paper gives an introduction to the theory of partial differential equations by discussing the main examples (Laplace's equation, Poisson's equation, transport equation and wave equation) and their applications.

### Potential students

This paper is particularly relevant for students majoring in mathematics, statistics, zoology, economics, design or any other field where the natural world is being modelled by differential equations.

### Prerequisites

MATH 202 (Linear Algebra), MATH 203 (Calculus of Several Variables), COMO 204 (Ordinary Differential Equations).

### Course Outline

The paper will cover the following topics:

- Laplace's equation and Poisson's equation (harmonic functions, mean value theorem for harmonic functions, maximum principle, Green's function, boundary value problem)
- The transport equation (initial value problem, characteristics)
- The wave equation (d'Alembert formula, energy methods, domain of dependence, finite propagation speed, initial boundary value problem)
- Non-linear first order PDE (characteristics, shocks)

### Lecturer

Dr Florian Beyer, room 218.

### Lectures

Tuesday 11-12

Thursday 9-10

Alternating Fridays 10-11

Rooms TBA.

### Tutorials

Tuesday 3-4

Room TBA.

### Office hours

TBA

### Internal Assessment

There will be five marked assignment, five unmarked assignments and a midterm test.

### Literature

Lecture Notes: Will be made available chapter by chapter during the semester on the resource webpage. These lecture notes are the main reference for this paper.

Book: *Partial differential equations* / Lawrence C. Evans (soon on course reserve in the library).

A good book on ordinary differential equations can be downloaded here for free (if you are on the university network).

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = 0.2A + 0.2T + 0.6E**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the Tests 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:

**Plagiarism**

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**

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**

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**

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.