Dimensional Analysis and Scaling
Dimensional and dimensionless quantities, Buckingham Pi Theorem, characteristic scales, multiple scales.

Perturbation Methods
Regular perturbations, singular perturbations, inner and outer approximations, matched asymptotics.

Stability and Bifurcation
Stability and phase plane analysis, bifurcating solutions.

Calculus of Variations
Functionals, Gateaux variation, Euler-Lagrange equation, Lagrange multipliers.

Partial Differential Equations
Initial-value and initial-boundary-value problems, classification, conservation laws, constitutive equations, Green's functions, linear and nonlinear waves, Burgers' equation.

Mathematical Modeling
Topics selected from Newton's second law, conservation laws, population dynamics, chemical reactions, oscillatory motion, diffusion equations, mathematical biology, wave motion.

Reference:

Applied Mathematics, 2nd Edition, J. David Logan, John Wiley & Sons.