Undergraduate Course: Advanced Methods of Applied Mathematics (MATH10086)
|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
|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.
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 layers
Part 3: PDEs
(7) first order PDEs: quasilinear, characteristics, shocks.
(8) waves and diffusion
(9) Green's functions
(10) waves in space
(11) eigenvalue problems
Information for Visiting Students
|Pre-requisites||Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling.
|High Demand Course?
Course Delivery Information
|Academic year 2023/24, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 44,
Seminar/Tutorial Hours 10,
Summative Assessment Hours 3,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 20%, Examination 80%
||Hours & Minutes
|Main Exam Diet S2 (April/May)||3:00|
On completion of this course, the student will be able to:
- Calculate the asymptotic expansion of integrals, including to higher order for simple integrals.
- Compute WKB approximations of second order ode's.
- Calculate the asymptotic solution of singularly perturbed ode's by the method of matched asymptotic expansions and boundary layers.
- Use the method of characteristics to solve first order nonlinear PDEs and the ability to fit shocks in such solutions.
- Solve the heat and wave equations using eigenfunction expansions, integral transforms and Green's functions.
|Books that could be helpful for this course are:|
J.P. Keener, Principles of Applied Mathematics, Transformation and Approximation, Addison-Wesley, Reading, Massachusetts (1988).
G.F. Carrier, M. Krook and C.E. Pearson, Functions of a Complex Variable, McGraw-Hill, New York (1966).
J. Kevorkian and J.D. Cole, Multiple Scale and Singular Perturbation Methods, Springer-Verlag (1996).
G.B. Whitham, Linear and Nonlinear Waves, J. Wiley and Sons, New York (1974).
J. Kevorkian, Partial Differential Equations, Brooks/Cole, Pacific Grove, California (1990).
R. Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, Prentice-Hall, Englewood Cliffs, New Jersey (1983).
|Graduate Attributes and Skills
|Course organiser||Prof Adri Olde-Daalhuis
Tel: (0131 6)50 5992
|Course secretary||Mrs Alison Fairgrieve
Tel: (0131 6)50 5045