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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 2017 (Monday, October 9 to Friday, December 15)

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.
Photo of Jitesh Gajjar
Alice Thompson
Email Alice.Thompson@manchester.ac.uk
Phone 0161-3068951
Photo of Alice Thompson


Students


Photo of Noura Alharthi
Noura Alharthi
(Loughborough)
Photo of Mounirah Areshi
Mounirah Areshi
(Loughborough)
Photo of Brennen Fagan
Brennen Fagan
(York)
Photo of Puneet Matharu
Puneet Matharu
(Manchester)
Photo of David Snee
David Snee
(Northumbria)


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


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Assessment



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