Advanced Mathematical Methods

For honours courses please contact the School Office.

Description

Perturbation techniques underpin almost all physical applications of applied mathematics. Examples range from boundary-layer theory in viscous fluid flow, the description of shock waves in compressible fluids, acoustics, optics, describing the orbits of planets in celestial mechanics, nonlinear oscillations and chaotic dynamics.

Objective

The aim of this course is to present students with a systematic account of modern perturbation methods and demonstrate by example how they can be applied to the solution of differential equations. At the end of this subject, students will be equipped with all the tools necessary to analyse differential equations arising in a broad range of physically motivated problems.

Content

Introduction: Asymptotic expansions, algebraic equations. Asymptotic evaluation of integrals: Laplace's method. The method of stationary phase. The method of steepest descent. Stokes phenomena. Boundary value problems: Boundary-layer theory: transition layers. Method of strained coordinates. Boundary layer theory for partial differential equations. WKB method. Evolution equations: Regular perturbation methods. Poincare-Lindstedt method. The method of multiple scales. Averaging (if time permits).

Jim Denier Lecturer for this course

Delivery

2 one-hour lectures per week plus tutorials as required.

Assessment

Three hour examination (85%) and three written assignments (15%).

Graduate attributes

• Applying mathematical knowledge (1)
• Interpreting data and drawing conclusions (2)
• Formulating mathematical problems (3)
• Problem solving skills (4)
• Flexibility in working environment (5)
• Team participation and leadership (7)
• Self-directed study (8)
• Career development (9)

Linkage past

No past linkages have been noted.

Linkage present

No present linkages have been noted.

Linkage future

This course is not recorded as prequisite for other courses.

Restrictions

Assumed knowledge: Differential Equations (as obtained through, for example, Differential Equations and Fourier Series) and also some of Differential Equations III. The section on asymptotic expansions of integrals will require some background knowledge of Complex Analysis. Handouts providing this background will be distributed.

Recommended text

• Asymptotic Analysis, J. D. Murray, Springer-Verlag (1984)
• Perturbation Methods, E. J. Hinch, C.U.P. (1991).
• Advanced mathematical methods for scientists and engineers, C.M. Bender and S.A. Orszag, McGraw-Hill, (1978).
• Perturbation methods in applied mathematics, J. Kevorkian and J. D. Cole, Springer-Verlag (1981).
• Perturbation methods, A. Nayfeh, J. Wiley (1973).
• Multiple scale and perturbation methods, J. Kevorkian and J. D. Cole, Springer-Verlag (1996).