MASTER OF SCIENCE DEGREE IN APPLIED MATHEMATICAL MODELLING (MAMM)

PROGRAMME OVERVIEW

To develop knowledge, skills and competencies in the field of Applied Mathematics and Modelling relevant to various employment capabilities and careers in the world of work and society. To equip learners with research and analytical skills which can be used to solve real world problems. 

ENTRY REQUIREMENTS

Normal Entry: Prospective students must hold a good first Degree in Applied Mathematics/Mathematics, Applied Statistics/Statistics, Computer Science, Operations Research or any other relevant mathematical discipline from any university recognized by Midlands State University.

CAREER OPPORTUNITIES AND FURTHER EDUCATION

Employability: Research scientists in research institutions. Data analysts, software developers, risk analysts in financial institutions, actuarial analysts in insurance companies and epidemiologists in medical laboratories and lecturers in tertiary institutions.

Further Studies: Doctoral studies in Mathematical modelling, Epidemiology and/or any interdisciplinary programmes related to the modules offered in the programme.

PROGRAMME STRUCTURE

Summary of Modules arranged in logical sequence, and allocation of Notional Hours and Credits
Module name	Minimum body of knowledge and skills	Credits
Level One		200
Semester 1
MAMM8301 Introduction to Mathematical Modelling	Y	24
MAMM8302 Numerical Solution of Ordinary Differential Equations	Y	24
MAMM8303 Stochastic Probability Spaces	Y	24
Total number of credits		72
Semester 11
Students do all modules
MAMM8304 Advanced Dynamical Systems	Y	24
MAMM8305 Advanced Functional Analysis	Y	24
MAMM8306 Financial Mathematics (pre-requisite is Stochastic Differential Equations)	Y	24
		72
Level Two
Semester 1
Electives (Students select at least four modules)
MAMM8307 Industrial Statistics		24
MAMM8308 Mathematical Epidemiology		24
MAMM8309 Continuum Mechanics		24
MAMM8310 Integral Equations		24
MAMM8311 Variational Calculus		24
MAMM8312 Forecasting		24
MAMM8313 Advanced Fluid Dynamics		24
MAMM8314 Perturbation Methods		24
MAMM8315 Numerical Solution of Partial Differential Equations  (pre-requisite is Numerical Solution of Ordinary Differential Equations)		24
The choice of electives will be offered subject to staff availability.
Semester 11
MAMM8370 Dissertation	96

 

SYNOPSES 

(For all the 80% Modules Threshold. NB: Synopses are very central in that these are summaries of the key concepts to be taught in each module.

MAMM8301 Introduction to Mathematical Modelling: 

General principles of mathematical modelling and modelling skills needed for abstraction, idealisation, identification of important factors such as variables and parameters. Students will study case studies from the following areas: 

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 and subtle engineering of special compounds. 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. Models will be formulated 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 and braking functions. The train itself is a dynamical system with a lot of vibrations and other phenomena. Analysis of traffic flow, scheduling, congestion effects, planning timetables and derivation of operational characteristics. 

           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; biochemical processes, mechanical handling of special   

            sorts of fluids and raw materials. The control of microbial processes, 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 using probabilistic and combinatorial methods.

Case 6: Climate modelling: Basic model equations, Global climate models and regional climate models.

MAMM8302 Numerical Solutions of Ordinary Differential Equations:

Introduction. 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. 

MAMM8303 Stochastic 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. Introduction to jump diffusions.  Differential Equations.

     MAMM8304 Advanced Dynamical systems:

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.

MAMM8305 Advanced Functional Analysis:

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. Lebesgue Integral and Lp spaces, with special emphasis on the case p = 2.

MAMM8306 Financial Mathematics:

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). Levy processes in finance.

MAMM8307 Industrial statistics: 

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. Reliability.

MAMM8308 Introduction to Mathematical Epidemiology: 

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 and SIERS. Epidemiological measures and their relationship to disease transmission models. The reproductive number. Use of models to plan clinical trials. Modelling of sexual, waterborne and vector borne transmitted diseases. 

Immunological Modelling: focusing on modelling the pathogenesis of infectious diseases. The interaction of humans and pathogens. 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 will be given higher priority.

MAMM8370 Dissertation:

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.

MAMM8309 Continuum Mechanics:

Rigid deformable bodies. Concept of stress. Deformation and kinetics. Balance equations. Constitutive equations. Examples of complex material. Solution of problems in elasticity and viscoelasticity.

MAMM8310 Integral Equations:

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. Applications.

MAMM8311 Variational Calculus:

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. Applications.

MAMM8312 Forecasting:

Applications to business management. Multiple regression modelling. Binary choice models, multiple discrete choice models and limited dependent models. Time series analysis: ARIMA, ARMA and VARMA models. 

MAMM8313 Advanced Fluid Dynamics:

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. Applications to meteorology, aerodynamics and irrigation systems.

MAMM8314 Perturbation Methods:

Concept of asymptotic development. Elementary operations on asymptotic expansions. Equations containing a small parameter and or a region slightly perturbed from a regular figure. The solution in terms of the small perturbation parameter. Methods of regular perturbation. Examples of the possibility of non-uniform expansions. Methods of singular perturbation: Poincare-Lighthhill-Kuo, matched asymptotic expansion and multiple scales. All the methods will be illustrated by solving ordinary and partial differential equations.

MAMM8315 Numerical Solutions of Partial Differential Equations:

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 condition