Last updated 14/04/2021.

Course lecturer: Dr D A S Rees Department of Mechanical Engineering. Room 4E 2.54.

Telephone: (01225) 386775 (Office)
E-mail Address: or

Typeset notes covering both Fourier Series and Fourier Transforms may be found here .
A retypeset upgrade to the Fourier Transforms section (01/02/2021) may be found here .

Important: There will be a lot of integration by parts. Please refresh your knowledge on this
by going to my ME10304 Mathematics 1 website and checking out the 4th Integration video here .

Syllabus: My part of the syllabus for this semester includes the use of (i) Fourier Series and (ii) Fourier Transforms, to solve the three most common partial differential equations, namely, Fourier's equation, Laplace's equation and the wave equation.

The one for my part of the unit is scheduled for Wednesdays at 10:15.
This follows immediately after a LOIL for ME10305 Maths 2. Please be patient if I am slightly late. I have rescheduled the 1st years for 9:00.
The links for the Zoom calls for the LOILs are near the top of the ME20021 Moodle page.
   Feb 17th notes  Feb 17th video LOIL 1
   Feb 24th notes  Feb 24th video LOIL 2
   Mar 3rd notes  Mar 3rd video LOIL 3
   Mar 10th notes  Mar 10th video LOIL 4
   Mar 17th notes  Mar 17th video LOIL 5
   Mar 24th notes  Mar 24th video LOIL 6
   Apr 14th notes  Apr 14th video LOIL 7
   Apr 21st notes  Apr 21st video LOIL 8
   Direct link for LOIL 9 on April 28th at 10:15.

Lecture plan:
1.   Video 1 (42.42) Slides 1 (13/02/2021) Introduction to Separation of Variables for PDEs
2.   Video 2 (23.36) Slides 2 (19/02/2021) Further examples. Full problems with half-range Fourier Series
2b. Video 2b (22.43) Slides 2b (02/03/2021) Further examples. As video 2 but using the wave equation and many graphs!
3.   Video 3a (32.16) Video 3b (20.29) Slides 3 (27/02/2021) Fourier Cosine Series and Quarter-range Sine series (in video 3a). Solutions in finite domains (in video 3b).
4.   Video 4 (41.37) Slides 4 PDEs in polar coordinates
5.   Video 5 (34.20) Slides 5 Introduction to Fourier Transforms. 1. Definition and examples of transforms. Symmetries. Physical meaning.
6.   Video 6 (30.10) Slides 6 Introduction to Fourier Transforms. 2. The two Shift theorems. Symmetry theorem. Convolution theorem. An ODE example.
7.   Video 7a (30.10) Video 7b (27.10) Slides 7 Application to PDEs 1. Fourier's equation and Laplace's equation.
8.   Video 8 (47.28) Slides 8 Application to PDEs 2. FST and FCT. Introduction and example solutions.

WEEKLY PLAN FOR ME20021 2020/2021

  Mon 1st February DNJ     Wed 3rd February DNJ  
  Mon 8th February DNJ     Wed 10th February DNJ  
  Mon 15th February DNJ     Wed 17th February DASR  
  Mon 22nd February DNJ     Wed 24th February DASR  
  Mon 1st March DNJ     Wed 3rd March DASR  
  Mon 8th March DNJ     Wed 10th March DASR  
  Mon 15th March DNJ     Wed 17th March DASR  
  Mon 22nd March DNJ     Wed 24th March DASR  
  Mon 29th March Easter     Wed 31st March Easter  
  Mon 5th April Easter     Wed 7th April Easter  
  Mon 12th April DNJ     Wed 14th April DASR  
  Mon 19th April DNJ     Wed 21st April DASR  
  Mon 26th April No lecture     Wed 28th April No lecture  
  Mon 3rd May Revision week     Wed 5th May Revision week  

Separation of Variables and Fourier Series:
Sheet 1 (fundamental solutions): ( problem sheet ) ( solutions )
Sheet 2 (full problems I): ( problem sheet ) ( solutions )
Sheet 3 (full problems II): ( problem sheet ) ( solutions )
Sheet 4 (full problems III Polar coordinates): ( problem sheet ) ( solutions )
Sheet 5: (Fourier Transforms - Introductory bits): ( problem sheet ) ( solutions ) (This is now up to date!)
Sheet 6: (Fourier Transforms, Fourier Sine and Cosine Transforms) ( problem sheet ) ( solutions ) (This is now up to date!)

Helpful Handouts

Hyperbolic functions ( hyperbolics.pdf )
Some comments on the ODEs which arise in the separation of variables: ( comments )
The various types of Fourier Series (non exhaustive, believe it or not): ( handout )
An example of each of the Fourier Series showing convergence: ( handout )
Definition of the Fourier Transform and some of its properties: ( defns )
Introductory lecture on FTs: handout   slides
The Fourier Sine and Cosine Transforms handout: ( handout )
Here are some notes on Fourier himself, his life, work and death: ( notes )
Integration by Parts. Check out last year's notes at notes and the problem and solutions sheets which are retrievable from the Maths 1 webpage.

Past papers.
These may be found here. Or else my two questions are given below:

15/16   Outline solutions   Informal feedback document   Formal feedback document
16/17   Outline solutions   Informal feedback document   Formal feedback document
17/18   Outline solutions   Formal feedback document
18/19   Outline solutions   Formal feedback document
19/20   Outline solutions   Formal feedback document

There is a scanned version of the exam formula book here. (This is the new 2019/2020 version).

Supplementary notes.
These may be found here . They include (i) separation of variables and Fourier Series material, (ii) a whistle-stop tour of the standard textbook method for separation of variables, (iii) solution of PDEs using separation of variables and Fourier Series, (iv) definition and properties of the various Fourier Transforms, (v) solution of PDEs using Fourier Transforms. I've used some different examples in these notes compared with the lectures, and you'll also find some pretty pictures.