MA323-6-SP-CO:
Partial Differential Equations
2023/24
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Spring
Undergraduate: Level 6
Current
Monday 15 January 2024
Friday 22 March 2024
15
08 January 2024
Requisites for this module
MA201 and MA202
(none)
(none)
(none)
(none)
This module considers the properties of the most common first and second order PDEs (known as Equations of Mathematical Physics), the mathematical concepts behind them and analytical methods of solution for such equations.
In this module we shall concentrate on second order linear PDEs (known as Equations of Mathematical Physics): elliptic equations (Laplace's equation), parabolic equations (heat equations) and hyperbolic equations (wave equations). We will study also various topics from real and complex analysis used for solving such equations: the Sturm-Liouville problem, maximum principle for harmonic functions, Fourier series.
The aim of this module is:
- To provide a general understanding of the theory of linear PDEs and methods of solution the most important types of such equations arising in applications to Physics and Geometry.
By the end of this module, students will be expected to be able to:
- Use the method of characteristics to solve first-order partial differential equations.
- Classify a second order PDE as elliptic, parabolic or hyperbolic.
- Use separation of variables for suitable boundary value problems for second order linear PDE.
- Have a basic knowledge and understanding of the theory of Fourier series and how to apply them to obtain fundamental solutions of Sturm-Liouville problems.
The majority of physical processes and phenomena can be described by partial differential equations, i.e. equations involving partial derivatives (e.g. the Navier-Stokes equations of Fluid Dynamics, Maxwell's equations of Electromagnetism, Schrödinger equation in Quantum Mechanics, Einstein equations in General relativity).
The main difference from the case of ordinary differential equations is that there is no analogue of existence and uniqueness theorem for a generic PDE. Instead, there is a variety of initial and boundary value problems one can impose for finding solutions of wide classes of equations. In addition, some important classes of nonlinear PDEs are also considered.
Syllabus
- Linear Differential Operators.
- Method of Characteristics.
- The one-dimensional wave equation.
- The Sturm-Liouville problem.
- Fourier transforms and distributions (generalised functions).
- Parabolic equations.
- Hyperbolic equations.
- Elliptic equations.
equations
Teaching in the School will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.
The above list is indicative of the essential reading for the course.
The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.
Further reading can be obtained from this module's
reading list.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Test |
|
|
| Exam |
Main exam: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period)
|
| Exam |
Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment Period)
|
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Georgios Papamikos, email: g.papamikos@essex.ac.uk.
Dr Georgios Papamikos
g.papamikos@essex.ac.uk
Yes
No
No
Prof Stephen Langdon
Brunel University London
Professor
Available via Moodle
Of 32 hours, 30 (93.8%) hours available to students:
0 hours not recorded due to service coverage or fault;
2 hours not recorded due to opt-out by lecturer(s), module, or event type.
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