Course details
A core MAGIC course
Semester
- Autumn 2020
- Monday, October 5th to Friday, December 11th
Hours
- Live lecture hours
- 7
- Recorded lecture hours
- 13
- Total advised study hours
- 80
Timetable
- Thursdays
- 09:05 - 09:55 (UK)
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.
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
-
MS
Dr Mike Simon
- University
- University of Manchester
- Role
- Main contact
-
AT
Dr Alice Thompson
- University
- University of Manchester
Bibliography
Follow the link for a book to take you to the relevant Google Book Search page
You may be able to preview the book there and 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.
- Advanced Mathematical Methods (J.S.B. Gajjar, )
- Advanced Mathematical Methods for Scientists and Engineers (Bender and Orszag, book)
- Asymptotic Expansions of Integrals (Bleistein and Handelsman, book)
- Complex variables: introduction and applications (Ablowitz and Fokas, book)
- Perturbation methods (Hinch, book)
- Introduction to Fourier analysis and generalised functions (Lighthill, book)
Assessment
The assessment for this course will be released on Monday 11th January 2021 at 00:00 and is due in before Sunday 24th January 2021 at 23:59.
Assessment will be via a Take-Home exam. You will be given 4 or 5 questions to do (100 marks in total) and to pass the course you need 50%.
Please note that you are not registered for assessment on this course.
Files
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Lectures
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