# Applied Mathematics

## Courses Summary

The courses consist of a selection of topics drawn from differential equations, mathematical modelling, classical and continuum mechanics, control theory and optimization and image processing. In Mathematics of Finance Honours there is a core consisting of topics in mathematics of finance.

Our first year course gives an introduction to the ideas and methods of Computational and Applied Mathematics. In the second and third year, the four major areas of Computational and Applied mathematics are emphasised. These are Numerical Methods, Mechanics, Solutions of Differential Equations as well as systems of Differential Equations and Optimisation. In the fourth year there are several directions of study. These are Optimisation and Control, Dynamical Systems theory and Chaos, Classical and Continuum Mechanics, General Relativity, Numerical Analysis, Mathematics of Finance, Image Processing and Astrophysics. Supervision for Masters and Doctoral Degrees in these areas is also available in the department. Masters degrees may be obtained by coursework and research report; or by a dissertation. There is a very active graduate program in the department.

## Other Subjects

Computational and Applied Mathematicians must have a good understanding of Mathematics, Computer Science, Physics, Statistics and other subjects offered in the Mathematical and Physical Sciences. However, applied mathematicians must be versatile and willing to cooperate with experts in many other fields. Thus some knowledge of economics, biology or a branch of engineering is also often useful to and applied mathematician. Whatever the ancillary subjects, the computational and applied mathematician must be ever ready to combine his/her mathematical and computational expertise with an appreciation of the subject areas in which mathematical models arise.

### Computational and Applied Mathematics I

Course Coordinator: Mr Michael Mitchley

Tel: 011 717 6157

Office No: UG 3, Mathematical Sciences Building, West Campus

E-mail: Michael.Mitchley@wits.ac.za

#### First Year Topics: APPM1006

**Numerical Analysis I: Numerical Methods**

Errors and error analysis; Round off errors; Absolute and relative error; Significant figures; Rounding; Chopping; Floating point arithmetic; Vectors and matrices; Definitions: Special matrices; Matrix addition and subtraction; Scalar multiplication; Matrix and vector multiplication; Inverse of a square matrix; Norms of vectors and matrices; Determinants; Properties of matrices; Linear independence and rank; Systems of linear equations: Gaussian elimination, LU decomposition, Jacobi’s method, Gauss-Seidel method, SOR; Zero’s of non-linear equations: Bisection, Method of False Position, Fixed point methods, Secant method, Newton’s Fixed Point Method; One and two dimensional interpolation: Linear interpolation, Lagrangian interpolation, Newton-Gregory forward and backward differences, Hermite interpolation; Computer programming in MATLAB.

**Optimisation I: Linear Programming**

Assumptions of linear programming; mathematical formulation of LP-problems; Basic and basic feasible solution; Graphical Method; Simplex method; Method of penalty: use of artificial variable and the initial setting up; Big M method; Dual Linear Programming. Mechanics I: Newtonian Mechanics

Vector algebra and calculus: Kinematics; Straight-line motion; Motion in a plane; Dynamics: Newton s laws; Energy: Conservation of energy; Momentum: Conservation of momentum.

**Modelling I: Introduction to Mathematical and Computer Modelling**

Basic concepts and assumptions: The continuous function and its graph; Derivatives; Laws of differentiation; Higher order derivatives; Anti-derivatives; The Taylor and Maclaurin expansions of a function; The Euler Identities; Equating coefficients; Partial fractions; Ordinary differential equations: The concept of a differential equation; The concept of the solution of a differential equation; The order of a differential equation; The distinction between linear and non-linear differential equations; Homogeneity of differential equations; Classification of differential equations; Analytic solution of certain differential equations. Continuous Mathematical Models: The construction of models of population growth; The logistic model; Radio-active decay; The concept of half-life; Models of drug distribution in the body; The concept and application of compartments in modelling; Examples of economic, financial, biological and other models. Ordinary Difference Equations: The concept of a difference equation; The concept of the solution of a difference equation; The order of a difference equation; The distinction between linear and non-linear difference equations; Homogeneity of difference equations; Classification of difference equations; The analytic solution of certain difference equations; The numerical solution of difference equations; Discrete Mathematical Models: The construction of population growth models; Models of spread of disease; Biological and economic models; Computer simulation of the discrete models; Computer Simulation of Continuous Models: A brief introduction to numerical methods for the solution of differential equations; The computer simulation of some of the continuous models developed earlier in the course.

### Computational and Applied Mathematics II

Course Coordinator: Dr Byron Jacobs

Tel: 011 717 6137

Office No: UG 2, Mathematical Sciences Building, West Campus

E-mail: Byron.Jacobs@wits.ac.za

#### Second Year Topics: APPM2007

**Numerical Analysis II: Numerical Techniques**

Numerical Techniques Iterative methods for systems of nonlinear equations; Discrete Least squares approximation; Richardson extrapolation; Numerical differentiation; Numerical integration; Numerical solution of ordinary differential equations: Initial and boundary value problems; Eigenvalues and eigenvectors: Direct power method; Inverse power method; Inverse shifted power method; Computational assignments in MATLAB.

**Optimisation II: Nonlinear Programming**

Univariate optimization and line search; Multivariate unconstrained optimization; Steepest descent; Newton’s method; Quasi-Newton and direct search methods; Constrained optimization: Equality and inequality constraints; Active and passive constraints; Feasible direction and Lagrange multiplier; Computational assignments in MATLAB.

**Methods A: Applied Ordinary Differential and Difference Equations**

Introduction: Motivations for the study of differential and difference equations; Review of Integration Techniques: Differential of a function, Substitution, Integration by parts, Integrating rational functions, Common integrals, Scalar nth-order Ordinary Differential Equations; Preliminaries: Integration, Classification of first-order ordinary differential equations, General linear ordinary differential equations of order n, Equations explicitly independent of x, Equation explicitly independent of y; Difference Equations: Motivation (Loan from a bank); Definitions: Linear difference equations; Introduction to Systems: Some definitions; Higher-order equation; Motivation for algebraic notation; General system of equations; Linear algebra theory; Uncoupling of systems; Solutions of Linear Systems: Homogeneous constant coefficients; Fundamental matrix solution; Homogeneous variable coefficients; Properties of transition matrix; General non-homogeneous systems; Discrete systems.

View the course description

**Mechanics II:**

Constrained motion of a point particle: Generalized coordinates; Equations of constraint; Transformation equations; Virtual displacement; Generalized forces; Lagrange s equations of motion for a point particle; Generalization to motion of a rigid body: Centres of mass; Moments of inertia; Applications to small oscillations.

**Modelling II: Intermediate Mathematical and Computer Modelling**

Two-Dimensional Phase Diagrams: Definition of a phase trajectory and phase diagram; Technique of constructing a phase diagram from a first order mathematical formulation of a physical system; Interpretation of a phase diagram - both linear and nonlinear examples are considered; The concepts of stable, unstable, marginally stable and static equilibrium positions; Stability of linear and nonlinear differential and difference equations: Linearization of systems about an equilibrium point; Introduction to dimensional analysis; Introduction to stochastic modelling; Modelling of Deposits and Bond repayments; Two body problem; Computational assignments in MATLAB.

### Computational and Applied Mathematics III

Course Coordinator: Dr Rhameez Sheldon herbst

Tel: 011 717 6114

Office No: UG 4, Mathematical Sciences Building, West Campus

E-mail: Sheldon.Herbst@wits.ac.za

#### Third Year Topics: APPM3017

**Numerical Analysis III: Advanced Numerical Techniques**

Symmetric and Non-symmetric eigenvalue problem: Jacobi Method, Givens Method, Householder Method, QR Method, QR Method with shifting, QR Method with repeated and complex eigenvalues; Approximation theory: Orthogonal polynomials, Continuous least squares, Gram-Schmidt Method, Chebyshev polynomials, Padé approximation, Fourier series, Non periodic functions; Numerical techniques and theory for the solution of ordinary differential equations: Linear multistep methods, Adams Methods, Order and Stability; Numerical solution of partial differential equations: Elliptic Partial differential equations on regular regions. Computational assignments in MATLAB.

**Optimisation III: Nonlinear Programming and Techniques**

Nonlinear Programming; Unconstrained Nonlinear Optimization; Quasi Newtonian Methods; Conjugate Direction and Conjugate Gradient Methods; Constrained Nonlinear Optimization; Problems with Equality Constraints; Problems with Inequality Constraints; Game Theory; Zero Sum Games; Non-Zero Sum Games.

**Methods B: Methods of Applied Mathematics**

Power Series Solution to Variable Coefficient ODE’s; Frobenius Method; Separation of Variable Method for PDE’s; Eigenvalue Problems; Sturm-Liouville Theory; Canonical Heat Equation.

**Mechanics III: Continuum Mechanics**

Tensor Analysis and application; Balance Laws and Field Equations; Isotropic Linear Elastic Solids; Isotropic Newtonian Fluids.

**Modelling III: Advanced Mathematical and Computer Modelling**

Dimensional Analysis; Stability Analysis; Population Modelling; Problems in Industry; Modelling Problems in MATLAB and Mathematica.

**Optimal Control Theory III**

Controllability and Observability; Optimal Control; Dynamic Programming for Discrete-Time Problems; Dynamics Programming for Continuous-Time Problems.