F1.3YQ1 Asymptotic analysis

(Third year module given in the Autumn term.)

Introduction

Lecturer. Simon Malham, Room CM T.21, Mathematics Department.

Contact. email: simonm@ma.hw.ac.uk and tel: 0131 451 3254.

Lectures. Monday, Tuesday and Friday at 1:15pm.

Tutorials. One a week: Tuesday at 2:15pm.

Webpages. This course homepage where you can download lecture notes, further handouts, solutions to the exercises, Maple or Mathematica worksheets and other stuff is:

which can also be reached from the Mathematics Department's Homepage --> Teaching --> Information for students already on course --> Modules mainly taken by Mathematics and AM&S students.

Course aims and objectives. To introduce techniques of asymptotic approximation for the solution of algebraic and differential equations and for the evaluation of integrals.

Syllabus.

• Asymptotic analysis. Approximations using Taylor's theorem; order notation; approximate solution of algebraic and transcendental equations; asymptotic series; evaluation of integrals with a large or small parameter; Laplace's method; Watson's lemma; method of stationary phase; method of steepest descents. (18 lectures)
• Introduction to modelling. Use of conservation laws; derivation of some standard partial differential equations. (6 lectures)

Assessment. The continuous assessment consists of a one hour midterm exam counting for 15 percent of the final mark (date to be announced) and a two-hour final exam at the end of term which counts for 85 percent of the course mark.

There is a resit in August/September for the ordinary course. The resit assessment is purely on the basis of a two-hour exam.

Calculators. In the final exam you will only be allowed to use either the Casio fx-85WA or fx-85MS. This is a University regulation. Personally, I do not think you will need a calculator in the Asymptotic Analysis exam.

Contract. Students are expected to read the notes in this booklet before, during and after the lectures and tutorials. Lectures will act as a more formal forum for the lecturer to explain the ideas of the course and give alternative examples, whilst tutorials will take a less formal and more personal form. There are starred exercises on problem sheets 1--7 and students must attempt these. Mathematics is best learned through grappling with the underlying ideas presented in lectures and then tackling problems given in the exercises.

Hence try the exercises, and if you get stuck, ask the lecturer either after a lecture, during the tutorials or during the office hours. It is vital that you can solve problems proficiently. If you need help, then

Office hours. Once a week; we'll try to agree on these in the first week of term.

Attendance sheets. Students will be required to sign an attendance sheet with their initials in every lecture and tutorial. If any one student misses three consecutive such contact events, or more than one-third of them overall up until that date, then their personal mentor will be contacted.

Evaluations. At the end the course students will have an opportunity to fill out formal university evaluations on the course.

Books. The two main recommended books are Bender and Orszag and Hinch (see the bibliography of the main notes for details).

Announcements

• (1/10/07) Office hours: Time to be announced. They will be in my office CM T.21.
• (1/10/07) Monday and Friday lectures: Please note that informally I want to we move forward the Friday lecture at 1:15pm in CM S.01 to Tuesday.

Electronic resources

PDF files are recommended for printing. denotes files not yet available.

Syllabus for 2004 (from the official department module pages)

Lecture notes.

 Series and power series Asymptotic analysis Maple worksheet (.mws Maple file) Heat equation General polynomial and transcendental equations (non-examinable) Chemical reaction kinetics (non-examinable)

Problem sheets and solutions. The solutions will be made available as the course progresses.

 Topic Date due Solutions Taylor series and order notation 14th Oct Solutions 1 Regular perturbation problems 21st Oct Solutions 2 Singular perturbation problems 28th Oct Solutions 3 Asymptotic series 4th Nov Solutions 4 2nd part Laplace integrals 11th Nov Solutions 5 Method of stationary phase 18th Nov Solutions 6 Modelling 25th Nov Solutions 7

Movies. These are the matlab source files for the movies shown during the course. Download them and use them freely.

Exam papers. Hardcopies of solutions for the December 2001 -- December 2003 papers can be obtained from me in SRG04. We will discuss these past papers in week 9 during the usual lecture times.

Module timetable

This may change slightly as we progress so keep checking this webpage.

 Lecture Section Topic Oct 1st - Series revision Oct 2nd - Power series revision Oct 5th 1 Order notation Oct 8th 2.1 Regular perturbations (expansion method) Oct 9th 2.1 Regular perturbations (iteration method) Oct 12th 2.1 Regular perturbations (non-integral powers) Oct 15th 2.2 Singular perturbations Oct 16th 2.2 Singular perturbations (transcendental equation) Oct 19th 3.1 Asymptotic series Oct 22nd 3.2/3.3 Asymptotic expansions and Properties of AEs Oct 23rd 3.3 Properties of AEs (continued) Oct 26th 3.4 Asymptotic expansions of integrals Oct 29th 4/4.1 Laplace's method Oct 30th 4.1 Laplace's method and Stirling's formula Nov 2nd 4.2 Watson's lemma Nov 5th 5 Method of stationary phase Nov 6th 5 Method of stationary phase (continued) Nov 9th - Application Nov 12th - Mathematical modelling 1 Nov 13th - Mathematical modelling 2 Nov 16th - Mathematical modelling 3 Nov 19th - Mathematical modelling 4 Nov 20th - Mathematical modelling 5 Nov 23rd - Mathematical modelling 6 Nov 26th - Revision and past papers Nov 27th - Revision and past papers Nov 30th - Revision and past papers

This webpage and it's content was started on 5/10/2004.