Methods of applied mathematics
Undergraduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
Calculus of variations
- Euler–Lagrange equation and variational principles
- Brachistochrone and classical problems
- Constraints and Lagrange multipliers in variational problems
- Hamilton's principle and mechanics (introduction)
- Rayleigh–Ritz method (introduction)
Perturbation and asymptotic methods
- Regular perturbation expansions for algebraic equations
- Singular perturbations and boundary layers (introduction)
- Poincaré–Lindstedt method for weakly nonlinear oscillations
- Multiple scales (overview at undergraduate level)
- Asymptotic matching (introduction)
Integral equations and transforms
- Volterra and Fredholm integral equations (introduction)
- Green's functions for ODE boundary-value problems
- Transform methods for PDEs and integral representations
- Convolution and superposition principles
- Well-posedness and conditioning in applied problems
STEM / applied
Modeling and approximation
- WKB approximation for Schrödinger-type equations (introduction)
- Homogenization and averaging (conceptual overview)
- Stability analysis of equilibria in nonlinear models
- Numerical verification of asymptotic approximations
- Case studies from fluids, optics, and mathematical biology
Computational support
- Symbolic and numerical tools for perturbation series
- Bifurcation tracking in parameter-dependent models
- Sensitivity analysis and adjoint methods (introduction)
- Model reduction and dimensionless formulations
- Communication of assumptions in approximate methods
Notes
Topics reflect common methods of applied mathematics syllabi at US colleges and universities. Depth of perturbation theory and variational methods varies; graduate-level asymptotic courses cover more advanced material.