Course lecturer: Dr D A S Rees Department of Mechanical Engineering. Room 4E 2.54.
Telephone: (01225) 386775 (Office)
Email Address: D.A.S.Rees@bath.ac.uk or ensdasr@bath.ac.uk
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 .
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.
LOIL
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
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 halfrange 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 Quarterrange 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.



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
whistlestop 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.