MAM2046W - Applied Mathematics 2046

Course Outline:

This course will provide students with fundamental topics in Applied Mathematics. It consists of the following four modules: 2NA: NUMERICAL ANALYSIS (MAM2053S in EBE) 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 equations; existence and uniqueness of solutions. Linear equations of the n-th order; 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 (MAM2050S in EBE) 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. Chaos on a strange attractor, Lorentz map, logistic map, Hénon map, Lyapunov exponents. Fractals.