## 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 (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:

- 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 10-11 in room MAB21.

Thursday 9-10 in room MA241.

Alternating Fridays 10-11 in room MAB21.

### Tutorials

Thursday 3-4 in room MATH241.

### Office hours

Tuesday 3-5 in room 218 (Science III).

### Internal Assessment

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

The midterm test will take place on Friday, September 8, 10-11.

### 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

If:

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

all expressed out of 100, then your final mark F in this paper will be calculated according to this formula:

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

### Students must abide by the University’s Academic Integrity Policy

**Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **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 student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.