Course finder

<< return to browsing

Semester 2

Advanced Methods of Applied Mathematics (MATH10086)

Subject

Mathematics

College

SCE

Credits

20

Normal Year Taken

4

Delivery Session Year

2023/2024

Pre-requisites

Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling.

Course 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. Assessment is by assignments and final examination.

Course Description

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 approximation of solution of ODEs. The final part of the course focuses on PDEs. It introduces important techniques for the solution of several classes of linear PDEs (heat and wave equation) and nonlinear PDEs (first-order). The concept of shock waves, familiar from supersonic flight and fluid flow, is introduced. The fitting of shock and expansion waves into the solution of nonlinear hyperbolic pde's is dealt with in detail. Examples are drawn from traffic flow, supersonic fluid flow and erosion. Part 1: Asymptotics and integral transforms. (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, method of stationary phase.Part 2: ODEs (4) regular and singular perturbations(5) WKB approximations: first approximations(6) boundary value problems: boundary layersPart 3: PDEs (7) first order PDEs: quasilinear, characteristics, shocks.(8) waves and diffusion(9) Green's functions(10) waves in space(11) eigenvalue problems

Assessment Information

Written Exam 80%, Coursework 20%, Practical Exam 0%

Additional Assessment Information

Coursework 20%, Examination 80%

view the timetable and further details for this course