Undergraduate Course: Advanced Methods of Applied Mathematics (MATH10086)
Course Outline
| School | School of Mathematics |
College | College of Science and Engineering |
| Credit level (Normal year taken) | SCQF Level 10 (Year 4 Undergraduate) |
Availability | Available to all students |
| SCQF Credits | 20 |
ECTS Credits | 10 |
| Summary | Our understanding of the fundamental processes of the natural world is based to a large extent on ordinary and partial differential equations (ODEs and PDEs). This course extends the study of ODEs and PDEs started in earlier courses by introducing several ideas and techniques that enable the construction of explicit exact or approximate solutions.
Integral transforms often provide solutions through integral representations. The integrals involved are nontrivial and need to be approximated using asymptotic expansions that take advantage of large or small parameters. The first part of the course discusses both integral transform methods and asymptotic techniques for the approximation of the resulting integrals. A second part introduces asymptotic techniques for the direct approximations of solutions of ODEs. The final part of the course focuses on PDEs. It introduces important techniques for the solutions several classes of linear PDEs (heat, wave and Laplace equation) and nonlinear PDEs (first-order). |
| Course description |
Part 1: Asymptototics and integral transforms. (9h)
(1) integral transforms: Laplace and Fourier (partly revision)
(2) asymptotic expansion: definitions and notations.
(3) asymptotic methods for integrals: Watson¿s lemma, the Laplace method, saddle point method.
Part 2: ODEs (9h)
(4) regular and singular perturbations
(5) WKB approximations: first approximations
(6) boundary value problems: boundary layers
Part 3: PDEs (15h)
(7) first order PDEs: quasilinear, characteristics, shocks.
(8) waves and diffusions
(9) Green's functions
(10) waves in space
(11) eigenvalue problems
(12) nonlinear PDEs.
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Information for Visiting Students
| Pre-requisites | None |
Course Delivery Information
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| Academic year 2015/16, Available to all students (SV1)
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Quota: None |
| Course Start |
Semester 1 |
Timetable |
Timetable |
| Learning and Teaching activities (Further Info) |
Total Hours:
200
(
Lecture Hours 44,
Seminar/Tutorial Hours 10,
Summative Assessment Hours 3,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
139 )
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| Assessment (Further Info) |
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %
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| Additional Information (Assessment) |
Coursework 20%, Examination 80% |
| Feedback |
Not entered |
| Exam Information |
| Exam Diet |
Paper Name |
Hours & Minutes |
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| Main Exam Diet S2 (April/May) | | 3:00 | |
Learning Outcomes
1. Demonstrate the asymptotic property of approximations and distinguish regular and singular perturbation problems.
2. Find dominant balances in differential equations with a small parameter.
3. Compute leading-order approximations of integrals with a small parameter.
4. In simple cases, find complete asymptotic expansions of integrals.
5. Compute WKB approximations of second order ODEs.
6. Identify boundary layers in the solutions of differential equations, and apply matched asymptotics to derive leading-order approximations to the solutions.
7. Ability to use characteristics to analyse PDEs.
8. Ability to classify and analyse the three most classical PDEs.
9. Apply Rankine-Hugoniot conditions to find shock solutions of simple nonlinear PDEs.
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Reading List
Books that could be helpful for this course are:
Recommended :
C.M. Bender and S.A. Orszag, Advanced mathematical methods for scientists and engineers, Springer, 1999.
F.W.J. Olver, Asymptotics and Special Functions, Wellesley, 1997.
W.A. Strauss, Partial Differential Equations: an introduction, 2nd edition, Wiley, 2008. |
Additional Information
| Graduate Attributes and Skills |
Not entered |
| Keywords | AMAM |
Contacts
| Course organiser | Dr Noel Smyth
Tel: (0131 6)50 5080
Email: |
Course secretary | Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: |
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