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Department of Mathematics
MATH232 - Mathematical
Techniques
Unit Syllabus
Fourier Series and Integral
Transforms:
- Algebraic background to Fourier series.
- Fourier series, pointwise convergence,
Dirichlet's theorem, uniform convergence, Parseval's identity,
uniqueness.
- Gibbs phenomenon, differentiation and
integration of Fourier series.
- Fourier transforms, inverse Fourier
transforms, Plancherel's identity, convolution.
- Applications to partial differential
equations.
- Laplace transform, the convolution theorem
and the solution of ordinary differential equations.
- The z-transform, the convolution theorem
and the solution of difference equations.
Mathematical Modelling:
- Ordinary differential equations: mass
on a damped spring; RLC circuit; the pendulum; motion under friction;
parametric dependence; the effect of nonlinearities; predator-prey
models; illustration by MATLAB; phase plane and limit cycles.
- Partial differential equations: heat
flow along a rod and other diffusion models; solution by Fourier
methods; one-dimensional waves; potentials, Laplace's equation,
with application to fluid flow and gravitational field of the
earth.
- Maps: the logistic equation, a short
introduction to bifurcation and chaos.
Numerical Methods for Differential
and Integral Equations:
- Discussion without proof of uniqueness
of solution to a differential equation.
- Initial value problems and boundary
value problems.
- Finite difference methods for ordinary
differential equations.
- Finite difference methods for partial
differential equations.
- Use of MATLAB to find solutions.
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