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General


This course is part of the MAGIC core.

Description

This is a core applied module. The aim of the course is to pool together a number of advanced mathematical methods which students doing research (in applied mathematics) should know about. Students will be expected to do extensive reading from selected texts, as well as try out example problems to reinforce the material covered in lectures. A number of topics are suggested below and depending on time available, most will be covered. The course proceeds at a fairly fast pace.
Assessment The assessment for this module will be in the form of a take-home exam at the end of the course.

Recommended books:
  • Bender and Orsag, Advanced mathematical methods for scientists and engineers
  • Bleistan and Handlesman, Asymptotic expansions of integrals
  • Hinch, Perturbation methods
  • Ablowitz & Fokas Complex Variables, C.U.P.
  • Lighthill Generalised Functions, Dover paperback.

Semester

Autumn 2016 (Monday, October 3 to Friday, December 9)

Timetable

  • Mon 12:05 - 12:55
  • Thu 09:05 - 09:55

Prerequisites

It is assumed that students have done some real and complex analysis.

Syllabus

  • Advanced differential equations, series solution,classification of singularities. Properties near ordinary and regular singular points. Approximate behaviour near irregular singular points. Method of dominant balance. Airy, Gamma and Bessel functions.
  • Asymptotic methods. Boundary layer theory. Regular and singular perturbation problems. Uniform approximations. Interior layes. LG approximation, WKBJ method.
  • Generalised functions. Basic definitions and properties.
  • Revision of basic complex analysis. Laurent expansions. Singularities. Cauchy's Theorem. Residue calculus. Plemelj formuale.
  • Transform methods. Fourier transform. FT of generalised functions. Laplace Transform. Properties of Gamma function. Mellin Transform. Analytic continuation of Mellin transforms.
  • Asymptotic expansion of integrals. Laplace's method. Watson's Lemma. Method of stationary phase. Method of steepest descent. Estimation using Mellin transform technique.
  • Conformal mapping. Riemann-Hilbert problems.

Lecturers


Jitesh Gajjar (main contact)
Email j.gajjar@manchester.ac.uk
Phone (0161) 2755895
Interests Stability theory, theoretical and computational fluid dynamics, R>>1 flows.
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Photo of Jitesh Gajjar
Alice Thompson
Email Alice.Thompson@manchester.ac.uk
Phone 0161-3068951
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Photo of Alice Thompson


Students


Photo of Hani Alahmadi
Hani Alahmadi
(Keele)
Photo of Asmahan Alajyan
Asmahan Alajyan
(Leeds)
Photo of Saeed Alamry
Saeed Alamry
(Leicester)
Photo of Amal Alharbi
Amal Alharbi
(Loughborough)
Photo of Hassan Alkhayuon
Hassan Alkhayuon
(Exeter)
Photo of Walid Almalki
Walid Almalki
(Cardiff)
Photo of Ahmed Alshehri
Ahmed Alshehri
(Cardiff)
Photo of Ali Althobaiti
Ali Althobaiti
(Keele)
Photo of Ahmed Alzaidi
Ahmed Alzaidi
(Keele)
Photo of Jinrong Bao
Jinrong Bao
(Loughborough)
Photo of Adam Barker
Adam Barker
(Reading)
Photo of Jacob Brooks
Jacob Brooks
(Surrey)
Photo of Matthew Charles
Matthew Charles
(East Anglia)
Photo of Junbin Chen
Junbin Chen
(Durham)
Photo of Michael Foskett
Michael Foskett
(Surrey)
Photo of Sara Frecentese
Sara Frecentese
(Liverpool)
Photo of Jordan Gill
Jordan Gill
(Southampton)
Photo of Maha Helmi
Maha Helmi
(Keele)
Photo of Yuija Liu
Yuija Liu
(Loughborough)
Photo of Georgina Long
Georgina Long
(Exeter)
Photo of Xiao Ma
Xiao Ma
(Loughborough)
Photo of Daniel Miller
Daniel Miller
(Exeter)
Photo of Thomas Morley
Thomas Morley
(Sheffield)
Photo of Marianthi Moschou
Marianthi Moschou
(Manchester)
Photo of Chinedu Nwaigwe
Chinedu Nwaigwe
(*External)
Photo of Joseph Oloo
Joseph Oloo
(Keele)
Photo of Antonios Parasyris
Antonios Parasyris
(Loughborough)
Photo of Jake Percival
Jake Percival
(Sheffield)
Photo of Fabio Peruzzo
Fabio Peruzzo
(Leeds)
Photo of Courtney Quinn
Courtney Quinn
(Exeter)
Photo of Aleksandra Ross
Aleksandra Ross
(Sussex)
Photo of Joe Siddons
Joe Siddons
(Liverpool)
Photo of Thomas Stratton
Thomas Stratton
(Sheffield)
Photo of Leyla Sultanova
Leyla Sultanova
(Keele)
Photo of William Thomson
William Thomson
(Birmingham)
Photo of Morgan Tudball
Morgan Tudball
(Loughborough)
Photo of Evan Turnill
Evan Turnill
(Southampton)
Photo of Robert West
Robert West
(Leeds)
Photo of Hanyu Yin
Hanyu Yin
(Loughborough)
Photo of Adam Yorkston
Adam Yorkston
(East Anglia)


Bibliography


Advanced Mathematical MethodsJ.S.B. Gajjar
Advanced Mathematical Methods for Scientists and EngineersBender and Orszag
Asymptotic Expansions of IntegralsBleistein and Handelsman
Complex variables: introduction and applicationsAblowitz and Fokas
Perturbation methodsHinch
Introduction to Fourier analysis and generalised functionsLighthill


Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

Assessment



There will be 5 questions in the Take Home paper. You need to obtain 50% to pass the course. The paper will be released in early January 2017 and you should upload your solutions via the MAGIC website before the due deadline of midnight 22nd January 2017. Please use a scanner or photocopier to generate a pdf file which can be uploaded.

MAGIC022 Exam paper

Files:Exam paper
Released: Monday 9 January 2017 (44.6 days ago)
Deadline: Sunday 22 January 2017 (30.6 days ago)
Instructions:

Please answer any 5 questions. Each question is 20 marks and to pass you need 50%.



Recorded Lectures


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