Authorised by Academic Registrar, April 1996
Objectives The objectives of this subject are for students to develop skill in using the Laplace transform to solve linear, ordinary differential equations and in finding the Fourier series of a periodic function and using its convergence properties; be able to employ any of a suite of numerical approximation techniques; to solve a nonlinear equation in a single variable; find an interpolating polynomial, using limited data, solve a system of linear equations; estimate the dominant eigenvalue of a matrix; estimate the value of a definite integral; solve a nonlinear differential equation, or find a `best fit' representation of a function; have a good idea of the likely numerical imprecision inherent in the results, and know (in some instances) ways to reduce such inaccuracies.
Synopsis This subject is designed to introduce techniques and applications of Laplace transforms and Fourier series, and to introduce suitable methods for the numerical problems and approximations most commonly encountered in engineering and science, and to examine their relative advantages, their likely accuracy, and their computational efficiencies. Topics include Laplace transforms (properties, applications including ordinary differential equations and linear systems generally; convolution, Heaviside and Dirac functions, inversion); Fourier series including harmonic analysis and convergence properties; floating point representation of numbers; propagation of errors; solution of non-linear equations in a single variable; interpolation; numerical solution of systems of linear equations by direct and indirect methods; least squares fitting; numerical integration including the Romberg algorithm for accelerated convergence; solution of ordinary differential equations in both initial value and boundary value problems; examination of computational efficiency; rates for convergence; error analysis; operation counts and computing time; introduction to suitable software packages for numerical approximations. On-campus students are offered lectures (one hour per week in first semester, two hours per week in second semester) supplemented by study guides and assignment work, and an optional weekly tutorial. For distance education students there are study guides and sssignment work, and compulsory participation in a residential school.
Assessment Assignments: 40% + Examination: 60%