Master of Science in Applied Mathematical Modelling
- 2 YearsCourse Duration
- PostgraduateSkill level
Purpose of the Programme
To develop knowledge, skills and competences in the field of Applied Mathematical Modelling relevant to various employment capabilities and careers in the world of work and society. To prepare students for further studies and lifelong learning in Applied Mathematical Modelling.
Areas of Study:
Mathematical Modelling, Numerical solutions of Differential Equations, Stochastic Differential Equations, Financial Mathematics, Functional analysis, Dynamical Systems, Mathematical Epidemiology, Forecasting, Fluid Mechanics.
Use of mathematical techniques and models to obtain practical solutions to concrete problems.
Research, teaching and learning are professionally oriented and focused on real life problems.
Solving problems from many branches of science, engineering, information technology and commerce.
The minimum entry requirement shall be an Honours Degree in Mathematics with at least a Lower Second Class Division. Where an applicant holds an equivalent degree, Senate through the recommendations of the Department and Faculty of Applied Science shall make the final decision on the application.The Programme runs over a period of four years.
Teaching and Learning Methods:
Lectures, tutorials, computer laboratory classes, seminars, group work, industrial visits, industrial attachment, research project, individual independent study.
Written and oral examinations, tests, seminar presentations, industrial attachment report, final year research project report, continuous assessments.
1. Multidisciplinary: Ability to define and solve problems from multiple academic disciplines
2. Quantitative and innovative reasoning: Capability to draw on big data and use analytics for informed decision making and strive to seek new ways of doing things
3. Communication skills: Ability to communicate effectively and to present information orally and in writing and using ICTs to both expert and non-expert audiences
4. Analysis and synthesis: Capacity for analysis using mathematical methods and synthesis using logical arguments and proven facts
5. Ethical commitment: Professional integrity and awareness of impact of applied mathematical modelling on society and the environment
6. Entrepreneurial skills: Capability to identify and create new business ventures based on knowledge and new thinking paradigms
1. Deep knowledge: Ability to analyse data in terms of underlying principles and knowledge and by means of appropriate mathematical methods
2. Production skills: Ability to formulate and use mathematical models to better understand the real world for sustainable development
3. Technology development skills: Ability to develop new technologies in applied mathematical modelling with a view to enhance production efficiencies and outputs in industry
4. Problem-solving skills: Ability to solve a wide range of problems in the sciences, technology and industry, and other fields; by identifying their fundamental aspects and using both theoretical and practical methods
5. Analytical and computational skills: Ability to use data to analyse various phenomena and technological issues using appropriate computer packages.
Intended Learning Outcomes
1. Ability to approach problems in an analytical and rigorous way, formulating theories and applying them to solve problems in problems in the sciences, technology and industry, and other fields;
2. Ability to apply mathematical techniques to problems arising from the planning, monitoring and management of large-scale systems such as health service, communication, energy distribution and transportation systems.
3. Ability to analyse and interpret data, finding patterns and drawing conclusions to support and improve business decisions;
4. Ability to develop mathematical and statistical models
5. Ability to breakdown a complex system into simple and understandable models
6. Ability to design and conduct observational and experimental studies
7. Ability to demonstrate knowledge and understanding of fundamental concepts in areas of applied mathematical modelling
8. Ability to use mathematical and statistical packages to model and solve problems in applied mathematical modelling
9. Ability to deal with abstract concepts and to think logically
10. Ability to present mathematical arguments and conclusions with accuracy and clarity
11. Ability to identify problems in industry and the community and develop appropriate solutions
12. Develop mathematical models to solve current practical problems
13. Communicate effectively and present information methodically and accurately using multi-media
This programme is regulated by the Faculty of Applied Science and the General University Academic Regulations for Postgraduate Masters Degrees by Module-work.
The programme shall run for 18 months full-time and when offered on Block-Release, it shall run over 24 months.
STRUCTURE OF THE PROGRAMME
A student in Part I shall register for four taught modules in each semester. In Part II (which consists of one Semester), a student shall register for a Project module leading to a Masters Thesis which shall be submitted to the Department at least a month before the end of the Semester in Part II.
Block –Release Programme
A student registered on the Block – Release Programme shall register for four taught modules per block whilst in Part I. In Part II a student shall register for a Project Module that shall commence at the beginning of that Part. The Project Module shall run over two Block periods of six months each and the Project report shall normally be submitted to the Department at least a month before the end of Part II.
A student shall be allowed to proceed to register for a Project Module if he/she has successfully completed all the taught modules.
A student who is credited with all eight taught modules, but has fails to successfully complete the dissertation may be awarded a Postgraduate Diploma in Applied Mathematical Modelling.
Each module shall be assessed at the end of the semester through a written examination. Continuous assessment for the individual modules shall contribute 25% while the written examination shall contribute 75% unless otherwise stipulated as shown in 5.3. The Project Module shall be examined by dissertation and the student shall be required to give a defence to the authenticity of his/her project work before a Panel of Examiners. The dissertation shall normally be submitted for marking, one month before the end of the programme. The taught modules shall contribute 60% while the dissertation shall contribute 40% to the final overall mark.
A student shall be required to earn a total of 340 credits to be awarded the degree.
Modules Assessed at 50% Module-work
Module Examination Module-work
SMA5191 50 50
SMA5221 50 50
PROCEED AND REPEATING OF MODULES
A student may be allowed to repeat Part I if he / she has passed the number of modules as stipulated and allowed by the Faculty and General Academic Regulations. He / she may be allowed to proceed to Part II if he/she has successfully completed Part I.
A student who fails the project with a mark of at least 40% may be allowed to re-submit the project only once at a later date, normally within three months of notification of the result. The maximum mark for such work shall be 50%.
A student who fails a Project Report with less than 40% or after re-submitting the project has failed to satisfy the examiners, may be allowed to repeat the project or opt for the award of a Postgraduate Diploma in Mathematical Modelling. In repeating the project, a completely new project work shall be undertaken. A repeat of a Project shall be allowed only once.
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.
Careers in the retail and manufacturing industry; banking, finance and insurance industry; research institutions; non-governmental organisations; positions in academia, data mining
PhD in Applied Mathematics or in interdisciplinary programmes related to Applied Mathematics