Applied Mathematics 2046

Course outline:
The course consists of four 30-lecture modules. Modules 2OD and 2ND are offered in the first semester and modules 2BP and 2NA in the second semester. The aim of this course is to introduce the student to a selection of fundamental topics in Applied Mathematics. The syllabus covers the
following topics:
2NA NUMERICAL ANALYSIS (coded as MAM2053S for Engineering students)
Solutions to non-linear equations and rates of convergence. Direct and iterative methods for solving linear systems, pivoting strategies, matrix factorization, norms, conditioning. Solutions to initial value problems including higher order ordinary differential equations. Interpolation and approximation theory, splines, discrete and continuous least squares. Numerical differentiation and integration. Error analysis and control.
2OD ORDINARY DIFFERENTIAL EQUATIONS
First order linear and nonlinear equations; existence and uniqueness of solutions. Linear equations of the n-th order and systems of n linear first order equations. Nonhomogeneous linear equations and systems; variation of parameters; qualitative theory of nonlinear equations; phase plane analysis; externally and parametrically driven oscillators; resonances; application to the theory of nonlinear vibrations. Calculus of variations.
2BP BOUNDARY-VALUE PROBLEMS (coded as MAM2050S for Engineering students)
Boundary-value problems, Sturm-Liouville problems, Green's function. Variational calculus, applications to Lagrangean and Hamiltonian mechanics. Diffusion, Laplace's and wave equation. Solution by separation of variables.
2ND NONLINEAR DYNAMICS
Fixed points, bifurcations, phase portraits. Conservative and reversible systems. Index theory, Poincáre-Bendixson theorem, Liénard systems, relaxation oscillators. Hopf bifurcations, quasiperiodicity and Poincaré maps. Applications to oscillating chemical reactions and Josephson junctions. Chaos on a strange attractor, Lorentz map, logistic map, Hénon map, Lyapunov exponents. Fractals.