Advanced Applied Mathematics

Variational, spectral, nonlinear and asymptotic methods. Research-level tools used across engineering: calculus of variations and Hamilton's principle, Sturm–Liouville eigenproblems and Green's functions, nonlinear dynamics, bifurcations and chaos, and perturbation/asymptotic methods.

• Calculus of Variations — The calculus of variations finds functions (not numbers) that minimise an integral — the shortest path, the fastest slide, the shape of a hanging cable, the path of light. Learn functionals, the first variation, the Euler–Lagrange equation, the Beltrami identity, constraints via Lagrange multipliers, natural boundary conditions, and the link to Hamilton's principle and FEM.
• Sturm–Liouville Theory & Green's Functions — Sturm–Liouville theory explains why separation of variables works: self-adjoint boundary-value problems have real eigenvalues and orthogonal eigenfunctions that form a complete basis. Green's functions give the response to a point source, turning any forcing into an integral. Learn both, with vibrating strings, heat conduction and beam deflection examples and a NumPy eigenvalue check.
• Nonlinear Dynamics, Stability & Chaos — Dynamical systems theory studies how states evolve. Learn state space and phase portraits, equilibria and linearisation with the Jacobian, classification of fixed points (nodes, saddles, spirals, centres), Lyapunov stability, limit cycles, bifurcations, and chaos with sensitive dependence — using pendulums, oscillators and the logistic map.
• Perturbation & Asymptotic Methods — When a problem contains a small parameter ε, perturbation methods build approximate solutions as series in ε. Learn regular perturbation of algebraic equations and ODEs, order symbols (O, o, ~), singular perturbations and boundary layers, secular terms and multiple scales, and asymptotic expansions — the analytical toolkit behind boundary-layer theory, weakly nonlinear vibration and small-parameter engineering estimates.