Advanced Mathematical Methods

Course Coordinator: Dr Tony M T Chan, Grad Dip, MPhil (CUHK), PhD (CityU)

Course Developer: The Open University, UK, Course Team

Advanced mathematical methods are essential for all mathematical, science and engineering disciplines. This course teaches several mathematical methods and knowledge that can be used to solve ordinary and partial differential equations such as: Laplace’s equation, the wave equation and the diffusion equation, vector calculus and Fourier analysis. The study begins with an introduction to the subject of fluid-flow problems. Through these problems the course will develop students’ modelling skills and teach students to implement relevant methods for a simple flow problem.

MATH S333 is one of the higher-level courses for the programmes in Mathematical Studies. This course is presented at 12-month intervals.

Advisory prerequisite(s)
Any student taking this course will need to have a firm understanding of mathematics. You are advised to have already studied MATH S221, or gained an equivalent level of mathematical maturity, before studying this course.

Aims
This course aims to:

• Extend the theory of differential equations, partial differential equations, vector field theory and Fourier analysis.
• Develop further the analytical and numerical analysis needed to describe various real-life applications.
• Enhance learners’ knowledge and understanding of some physical situations and real-life models associated with the diffusion and dispersion processes.
• Teach mathematical modelling and the formulation of models for a simple flow fluid problem.

Contents
The course covers the following topics:

Unit 1 Properties of a fluid

• Illustrates the basic equations and mathematical methods for computing the nature of fluid mechanics.
• Introduces some of the physical properties of fluids and the continuum model of a fluid.

Unit 2 Ordinary differential equations

• Extends the methods of solving ordinary differential equations on the boundary-value and eigenvalue problems.
• Uses the method of power-series for solving initial-value problems.

Unit 3 First-order partial differential equations

• Extends the chain rule to cover a change of variables for functions of two variables.
• Teaches how this leads to the method of characteristics for solving first-order partial differential equations.

Unit 4 Vector field theory

• Extends the line, surface and volume integrals through two important theorems: (1) Gauss’ theorem and (2) Stokes’ theorem.
• Formulates and applies the equation of mass continuity for a fluid in motion.

Unit 5 Second-order partial differential equations

• Investigates analytically the solutions of second-order partial differential equations.
• Solves the second-order partial differential equations that can be classified into elliptic; hyperbolic and parabolic.

Unit 6 Fourier series

• Applies Fourier series together with the separation of variables to represent the solutions of initial-boundary value problems involving the diffusion equation and the wave equation.
• Introduce the Sturm-Liouville theory and related problems.

Unit 7 Laplace’s equation

• Solves Laplace’s equation and apply it to physical models.
• Interpret the solution in the context of fluid flow problems.

Learning support
There will be six two-hour tutorials and three surgeries throughout the course.

Assessment
There will be two assignments (both assignments are required for the course) and a final examination. Students are required to submit assignments via the Online Learning Environment (OLE).

Online requirement
This course is supported by the Online Learning Environment (OLE). You can find the latest course information from the OLE. Through the OLE, you can communicate electronically with your tutor and the Course Coordinator as well as other students. To access the OLE, students will need to have access to the Internet. The use of the OLE is required for the study of this course and you can use it to submit assignments.

Equipment
Students will need access to a computer with an Internet connection and the ability of sound input and output. A device capable of playing audio and video CDs will also be required. A scientific calculator is also necessary.

Set book(s)
There are no set books for this course.

Student with disabilities or special educational needs
The audio and visual components of this course may cause difficulties for students with a hearing or visual impairment. You are encouraged to seek advice from the Course Coordinator before enrolling on the course.