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APPLIED MATHEMATICS

Major combinations:
NQF Level: 5: MAT1503, MAT1512 and any TWO of the following: APM1513, APM1514, APM1612, PHY1505
NQF Level: 6: APM2611, MAT1613, MAT2615 and at least two other APM modules on NQF Level: 6
NQF Level: 7: FIVE of the following: (a) APM3701 (b) APM3711 (c) APM3712 (d) APM3713 (e) APM3706 (f) MAT3707

Introduction to General Relativity - APM4804
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: General relativity is the relativistic theory of gravitation, and is widely regarded as one of the major achievements of 20th century physical science. The detailed syllabus is: Review of special relativity. The equivalence principle and the physical ideas that lead to general relativity. The Einstein field equations. The linearised field equations: Comparison with Newtonian theory, and gravitational waves. The Schwarzschild solution: Derivation and properties. Introduction to black holes. The Friedmann-Robertson-Walker solution.
Mathematics of Optimization Theory - APM4805
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: The concept of optimization, in its various forms, is a very fundamental one with an important role to play in various branches of mathematics and of course also in the application of mathematics in other disciplines such as economics and engineering. The infinite dimensional case of optimization is studied in the calculus of variations and in optimal control theory. This module presents the classical theory of optimization in the finite dimensional situation. The emphasis is on the development of the mathematical theory and techniques of optimization (convex analysis, Lagrange multiplier rules) rather than computational or numerical techniques for finding optimal points.
Applied Linear Algebra - APM1513
Under Graduate Degree Year module NQF level: 5 Credits: 12
Module presented in English
Co-requisite: COS1511, MAT1503
Purpose: To enable students to master and apply the following aspects of the numerical solution of systems of linear equations: the method of least squares; linear programming (simplex method); eigenvalues, eigenvectors, diagonalisation as well as some miscellaneous applications.
Riemannian Geometry and Tensor Calculus - APM4806
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: Vectors and tensors in general coordinate systems. Covariant differentiation. The Riemann curvature tensor and associated tensors. The Weyl tensor and conformal metrics. Lie derivatives. Description of hypersurfaces. This module may be taken independently of APM4804, but it has been set up to provide the necessary mathematical background for a proper study of general relativity. It is concerned with the description of an N-dimensional non-Euclidean space referred to arbitrary coordinates.
Mathematical Modelling - APM1514
Under Graduate Degree Year module NQF level: 5 Credits: 12
Module presented in English
Purpose: To enable students to demonstrate a basic understanding of solution, equilibrium points and stability of difference equations and firstorder differential equations; applications to population models; harvesting strategies; epidemics; economics and other situations; simple optimisation and applications.
Numerical Solutions to Partial Differential Equations - APM4808
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: Partial differential equations (PDE's) have formed a basis of many mathematical models of chemical, physical and biological problems. More recently, the use of PDE's has also extended to include the fields of economics and financial forecasting. In this module we study various finite difference methods for the numerical solution of these PDE's. The efficiency of these methods is then examined by means of theoretical analysis of their consistency, convergence and stability.
Mechanics II - APM1612
Under Graduate Degree Year module NQF level: 6 Credits: 12
Module presented in English
Pre-requisite: MAT1512 & PHY1505
Purpose: To enable students to demonstrate a basic understanding of definite integrals, line integrals and the vector product; dynamics of systems of particles and rigid bodies; in particular mass centres, moments of forces, moments of inertia and angular momentum.
Optimal Control - APM4809
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: Systems that evolve in time occur naturally in various fields such as dynamics, economics, medicine and ecology, and modelling the behaviour of such systems provides an important application of mathematics. These systems can be completely deterministic, but often it may be possible to control their behaviour through the application of external controls. The theory of optimal control is concerned with finding the controls which, at minimum cost, either direct the system along a given trajectory or enable it to reach a desired target state. This module introduces some of the basic tools of optimal control. Many applications from various fields are also included, to show how the various 'maximum principles' help us find the optimal controls in practice. Contents: Terminology and classifications of optimal control problems. Controllability and reachability. Linear time optimal problem. The Time Optimal Principle. The Pontryagin Maximum Principle. Linear equations with quadratic costs.
Differential Equations - APM2611
Under Graduate Degree Year module NQF level: 6 Credits: 12
Module presented in English Module presented online
Pre-requisite: MAT1512 Co-requisite: MAT1503, MAT1613
Purpose: To enable students to obtain knowledge of first-order ordinary differential equations, linear differential equations of higher order, series solutions of differential equations (method of Frobenius), Laplace transform and partial differential equations (only an introduction).
An Introduction to the Finite Element Method - APM4810
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: I want to introduce you to a module on Finite Elements. This module will develop the basic mathematical theory of the Finite Element Method (FEM).This method is the most widely used technique for engineering design and mathematical physics. In studying this module the student will obtain a clear knowledge of what the Finite Element Method is, how it works and how to use it to solve boundary-value problems. The Finite Element Method is a general technique for constructing approximate solutions to boundary-value problems. The method involves dividing the domain of the solution into a finite number of sub domains, the finite elements, and using variational concepts to construct an approximation of the solution over the collection of finite elements.
Numerical Methods I - APM2613
Under Graduate Degree Year module NQF level: 6 Credits: 12
Module presented in English Module presented online
Pre-requisite: COS1511 Co-requisite: MAT1512, MAT1503
Purpose: To enable students to understand and use numerical methods in solving scientific and mathematical problems that are difficult to solve analytically. It includes solutions of non-linear equations and systems of linear equations, interpolating polynomials, numerical integration and differentiation, and least-squares approximation.
Applied Functional Analysis - APM4811
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: Research in the solvability of PDE's leads us to a much wider scope. In the realization that alternative methods can be used for proving the existence and uniqueness of solutions of linear and nonlinear PDE's, a new research area in Applied Mathematics was introduced. The theory of Sobolev Spaces was developed, which turned out to be a suitable setting in which to apply ideas of functional analysis to glean information concerning PDE's. For this reason I want to introduce you to a module on Applied Functional Analysis.
Computer Algebra - APM2616
Under Graduate Degree Year module NQF level: 6 Credits: 12
Module presented in English Module presented online
Pre-requisite: COS1511 and APM1513 Co-requisite: MAT1512, MAT1503
Purpose: To give students an understanding of the power of modern computer algebra systems, and specifically to enable students to use computer algebra to solve analytically a variety of mathematical problems including the algebraic equations (both linear and nonlinear), differentiation, integration, differential equations, matrix manipulation, series expansions, and limits; and to represent mathematical functions graphically, 2D and 3D, and to produce mathematical reports.
Introduction to Mechanics of Fluids - APM4812
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: The purpose of this module is to make students aware of some fundamental aspects of fluid motion, including important fluid properties, regime flow, pressure variations in fluids at rest and in motion, fluid kinematics, methods of flow description and analysis and the Bernoulli equation. This module conveys the essential elements of kinematics, including Eulerian and Lagrangian mathematical description of flow phenomena, and indicates the vital relationship between the two views. The basic analysis methods generally used to solve or to begin solving fluid mechanics problems (linear motion and deformation, angular motion and deformation, conservation of mass, conservation of linear momentum, viscous flow) are also introduced. Emphasis is placed on understanding how flow phenomena are described mathematically and on when and how to use infinitesimal and finite control volumes. Important notions such as boundary layers, transition from laminar to turbulent flow will also be introduced.
Applied Dynamical Systems - APM2614
Under Graduate Degree Year module NQF level: 7 Credits: 12
Module presented in English Module presented online
Pre-requisite: MAT1503
Purpose: To enable students to master and apply fundamental aspects of discrete and continuous systems including linear systems; phase portraits: equilibrium points, stability, limit cycles; Liapunov stability; elementary control theory as well as applications to mechanics, ecology, economics and elsewhere.
Symmetry Analysis - APM4815
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: The primary purpose of the qualification is to provide Honours Bachelor of Science graduate level knowledge, specific skills and applied competence in Pure and Applied Mathematics based on groups and algebras at Honours level. The qualification is to provide graduates who can understand the constructive role designed to solve differential equations analytically, that are applicable in the sciences, and broadly in engineering and technology in the local, regional and global contexts. Successful learners should have the ability to access and evaluate scientific information; competence in applying knowledge through basic research methods and practice. Another purpose is that learners would acquire specific mathematical skills and applied competencies leading to continued personal intellectual growth, gainful economic activity and valuable contributions to society in science, engineering and technology.
Partial Differential Equations - APM3701
Under Graduate Degree Year module NQF level: 7 Credits: 12
Module presented in English Module presented online
Pre-requisite: APM2611 & MAT2615
Purpose: To introduce students to the following topics in partial differential equations; the equation of Laplace, the heat equation and the wave equation treated as typical examples of elliptic, parabolic and hyperbolic partial differential equations respectively, and methods of solution of the corresponding boundary value problems are also discussed.
Honours Research in Applied Mathematics - HRAPM82
Honours Year module NQF level: 8 Credits: 36
Module presented in English Module presented online
Purpose: The purpose of this module is to introduce the student to academic writing skills for mathematics using LATEX and BibTEX, the process of conducting mathematics literature searches and the preparation of mathematics research documents, and to use these skills to engage in a research project in mathematics, under supervision of an academic supervisor.
Ordinary Differential Equations - APM3706
Under Graduate Degree Year module NQF level: 7 Credits: 12
Module presented in English Module presented online
Pre-requisite: APM2611
Purpose: This module will be useful to students interested in developing basic skills in ordinary differential equations. Mathematical models of real-life situations often lead to linear systems of differential equations. Linear systems of differential equations arise for instance in the theory of electric circuits, in Economics and in Ecology, the branch of science in which the interaction between different species is investigated. Students interested in these areas will benefit from this module. Students credited with this module will have an understanding of the basic ideas of the many types of solutions for linear systems of ordinary differential equations.
Numerical Methods II - APM3711
Under Graduate Degree Year module NQF level: 7 Credits: 12
Module presented in English Module presented online
Pre-requisite: APM2613
Purpose: To equip students with numerical techniques for the approximate solution of initial and boundary value problems of differential equations and function approximations.
Mechanics and Calculus of Variations - APM3712
Under Graduate Degree Year module NQF level: 7 Credits: 12
Module presented in English Module presented online
Pre-requisite: APM2611 & MAT2615
Purpose: To enable students to demonstrate a basic understanding of generalised coordinates, Hamilton's principle, calculus of variations and the Euler-Lagrange equations, the problem of Lagrange and the isoperimetric problem, Hamilton-Jacobi theory and Poisson brackets, Equivalent Lagrangians, canonical transformations and Noether's theorem and application of the variational principles in mechanics.
Special Relativity and Riemannian Geometry - APM3713
Under Graduate Degree Year module NQF level: 7 Credits: 12
Module presented in English Module presented online
Pre-requisite: MAT2615
Purpose: To introduce students to special relativity and the basics of general relativity. Introductory geometry in non-Euclidian spaces and tensor algebra will also be covered.
Cosmology - APM4801
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: Cosmology is the study of the physical universe. The module first introduces properties of the visible universe, including concepts such as distance scales, redshift, isotropy and homogeneity. This is followed by a brief survey of the structure and evolution of galaxies and stars. An empirical basis is used to show that the physical universe in its entirety has structure and evolves. The module focuses mainly on big-bang models of the universe and gives a description of both Newtonian cosmology and general relativistic cosmology. The big-bang type of evolution of the universe is followed from its early stages, including neutrino decoupling and the radiation dominated era. This is then pursued through decoupling and the origin of the cosmic microwave background radiation, and into the matter dominated era. The module is concluded with a fairly introductory discussion of observational cosmology. The latter looks at a variety of cosmological observations, all using discreet sources of radiation, to test the validity of models. The module is aimed at students who majored in applied mathematics, physics, or astronomy.
Continuous Time Stochastic Processes - APM4802
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: The purpose of this module is for students to gain insight into the use of continuous time processes as a tool of applied mathematics.