SMA5111 Advanced Functional Analysis 25 Credits
The module looks at metric spaces; Definitions and examples; Rn, C[a,b]; Inequalities of Holder, Minkowski, Cauchy-Schwarz; Open and closed sets, neighbourhoods; Convergence, completeness; Contraction Mapping Theorem; Applications to linear systems, integrals equations, differential equations; Normed spaces; Definitions and examples; Banach space; Finite dimensional space; Compactness and Riesz Lemma; Linear operators and functionals; Dual space; Second dual; Reflexivity; Weak convergence; Hilbert spaces; Definitions and examples; Cauchy-Schwarz inequality, Pythagoras’s theorem; Orthogonal complements and direct sums; Orthonormal sets; Fourier series and orthogonal polynomials; Hilbert adjoint operator; Self-adjoint operators; Eigenvalues and eigenfunctions; Operators; General measure theory as well as Lebesgue Integral and Lp spaces with special emphasis on the case p = 2;

SMA5131 Continuum Mechanics 25 Credits
This module explores rigid deformable bodies; Concept of stress; Deformation and kinetics; Balance equations; Constitutive equations; Examples of complex material together with Solution of problems in elasticity and viscoelasticty;

SMA5141 Integral Equations 25 Credits
The module outlines iterative methods for linear systems; Initial and Boundary Value Problems for ODES; Methods for Fredholm Integral Equations of the second kind; Neumann series; Degenerate kernels; Quadrature methods; Expansion methods and Applications.

SMA5151 Variational Calculus 25 Credits
The module highlights calculus of Variations: Function of one variable and several variables, constrained extrema and Lagrange multipliers, Euler-Lagrange equations; Functions with higher-order derivatives and several dependent variables and independent variables as well as applications.

SMA 5161 Numerical Solutions Of Ordinary Differential Equations ` 25 Credits
The module is an introduction to Matrix Analysis; Solutions of system of linear and non-linear differential equations, Linear and Non-linear initial value problems, Existence and uniqueness of solutions; Dependence of solutions on initial conditions; Numerical methods: - Euler, Runge-kutta methods; Multistep methods and variable step-size methods – Predictor-corrector methods; Refining of the step size convergence; Convergence and stability; Boundary value problems; Shooting methods for linear and nonlinear problems; Finite difference methods for linear and non-linear problems; Raleigh-Ritz method; Applications: growth models and epidemiological models.

SMA5181 Stochastic Differential Equations 25 Credits
The module is about probability spaces; Random variables and stochastic processes, Ito integrals, Ito’s formula and martingale representation theorem; Stochastic differential equations; Diffusions, Boundary value problems; Optimal stopping; Stochastic control and an introduction to jump diffusions.

SMA5191 Introduction to Mathematical Modelling 25 Credits
The model looks at the general principles of mathematical modelling and modelling skills needed for abstraction, idealisation, identification of important factors such as variables and parameters. Case studies shall be chosen from the following list hence Students shall study case studies from the following case studies.
Case 1: Simulation modelling; Discrete event simulation; Systems dynamics; Simulation software; Sampling methods; Model testing and validation.
Case 2: Materials Science Modelling: Understand the micro-level molecular and sub-atomic effects, subtle engineering of special compounds etc; The behaviour of non-typical materials or new materials like semiconductors, polymer crystals, composite materials, piezoelectric materials, optically active compounds, optical fibres etc; create a multitude of research questions, some of which can be approached with mathematical models and models to design and control the manufacturing processes.
Case 3: Traffic and Transportation Modelling: Roads, railway networks and air traffic contain many challenges for modelling; In railway industry, mechanical models about the rail-wheel contact, explaining the phenomena of wear, slippage, braking functions etc; The train itself is a dynamical system with a lot of vibrations and other phenomena; Analysis of traffic flow; Scheduling, congestion effects, planning timetables, derivation of operational characteristics etc.
Case 4: Modelling in Food and Brewing Industry: Mathematics has to do with butter packages, lollipop ice-cream, beer cans and freezing of meatballs; The food and brewing industry means biochemical processes, mechanical handling of special sorts of fluids and raw materials; The control of microbial processes and production chain.
Case 5: Chemical Reactions and Processes Modelling: Chemical processes modelled on various scales; The spatial structures and dynamical properties of individual molecules, to understand chemical bonding mechanisms etc; The chemical reactions are modelled use of probabilistic and combinatorial methods.

SMA5211 Advanced Dynamical Systems 25 Credits
The module explains systems of differential equations; Two-dimensional linear and almost linear autonomous systems; Finite difference equations; Steady states and their stability; Stability of periodic orbits; Lyapunov methods; Bifurcation, one and two- dimensional systems; Discrete systems; Self-similarity and fractal geometry; Chaos detecting and route to chaos.

SMA5221 Forecasting 25 Credits
The module explores applications to business management; Multiple regression modelling; Binary choice models, multiple discrete choice models and limited dependent models as well as time series analysis: ARIMA, ARMA and VARMA models.

SMA5231 Advanced Fluid Dynamics 25 Credits
The module outlines the basic principles of fluid dynamics; Equations of continuity and motion; Dynamical similarity; Some solutions of viscous flow equations; Inviscid flow; Boundary layers; Instability and turbulent flows; Flow in rotating fluids; Geotropic flow, Ekman layer and Rossby waves; Stratified flow; Stratification and rotation.

SMA5241 Perturbation Methods 25 Credits
The module looks at the concept of asymptotic development; Elementary operations on asymptotic expansions; Equations containing a small parameter and or a region slightly perturbed from a regular figure; Solution in terms of the small perturbation parameter; Methods of regular perturbation; Examples of possibility of non-uniform expansions; Methods of singular perturbation: Poincare-Lighthhill-Kuo, matched asymptotic expansion and multiple scales. All the methods shall be illustrated by solving ordinary and partial differential equations.

SMA5251 Industrial Statistics 25 Credits
The module looks at principles of experimental design; Completely randomised designs; Randomised Block designs; Balanced incomplete Block designs; Latin square and crossover designs; Factorial designs; Fractional factorial designs; Response surface methodology; Nested designs; Split-plot designs; Repeated measures designs; Analysis of covariance; Quality control and reliability.

SMA5261 Numerical Solutions Of Partial Differential Equations 25 Credits
This module explores the elliptic Partial differential equations; Poisson Problem with Dirichlet, Neumann and Robin Boundary Conditions, finite difference method; Parabolic partial differential equations; Initial-boundary value problems, one-dimensional explicit and implicit methods, stability; First-order hyperbolic partial differential equations; Finite Element Methods and Variational Techniques: Introduction-functional, Green’s theorem, divergence theorem, Euler-Lagrange equations, mixed boundary conditions, functionals for differential problems; Approximation of solution-Ritz method; Variational and weak forms in Hilbert (Sobolev) spaces; Finite Element Methods: Review of elliptic and partial differential equations, Laplace, Poisson, biharmonic and Lame’s equations all with various types of boundary conditions; Lagrange basis functions; Applications and Fluid flow models, for example air quality modelling.

SMA5281 Financial Mathematics 25 Credits
The course is an introduction to financial derivatives, the Cox-Ross-Rubinstein model; Finite security markets; Market imperfections; The Black-Scholes model; Foreign market derivatives; American options; Exotic options; Continuous-time security markets; Arbitrages and equivalent Martingale measures; The one period model; Multi period models; The continuous model; Hedging and completeness; Self financing portfolios; Attainability of a claim; Complete markets; Ito representation theorem; Girsanov’s first theorem; Option pricing; European options; American options; The Black Scholes option pricing formula; Optimal portfolio and stochastic control; Stochastic control theory; The Hamilton-Jacobi-Bellman equation; Girsanov’s second theorem; Numerical analysis in finance (solving nonlinear partial differential equations arising in finance; Use of appropriate computer packages in finance e;g; Matlab) as well as levy processes in finance.

SMA5291 Introduction To Mathematical Epidemiology 25 Credits
The module looks at modelling Transmission Dynamics of Infectious Diseases: Basic concepts of epidemiological modelling; Epidemiological principles and concepts; Tools required to develop mathematical models to understand the transmission dynamics of infectious diseases and to evaluate potential control strategies. Topics to be covered include: history of mathematical epidemiology, Introduction to population modelling, Basic models of disease transmission, SI, SIS, SIR, SIRS, SIE, SIER, SIERS etc; Epidemiological measures and their relationship to disease transmission models; The reproductive number; Use of models to plan clinical trials as well as modelling of sexual, waterborne and vector borne transmitted diseases. The module also looks at immunological Modelling: focusing on modelling the pathogenesis of infectious diseases; The interaction of human and pathogen; The biochemical, pharmacological, immunological, and molecular biological understanding of how infectious agents and the human body interact; Development of models to study host susceptibility to particular pathogens, development of models to study host resistance to chronic or acute disease, development of models for basic studies of infectious organisms, as long as they are oriented toward understanding how the organism interacts with the host, development of models to study virulence factors, immune mechanisms, and genetic studies in the host and in the pathogens. Work on modelling pathogenesis of HIV, malaria, and tuberculosis shall be given higher priority.

SMA5010 Dissertation 140 Credits
The dissertations may be carried out on an individual basis. The dissertation normally involves work with some outside organization. The dissertations test students’ ability to organise, complete and report on a significant piece of Mathematical modelling.

 

Employability:
Careers in the retail and manufacturing industry; banking, finance and insurance industry; research institutions; non-governmental organisations; positions in academia, data mining

Further Studies:
PhD in Applied Mathematics or in interdisciplinary programmes related to Applied Mathematics