Partial differential equations
Undergraduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
First-order equations and classification
- Partial derivatives and notation; initial and boundary value problems
- First-order linear and quasilinear PDEs; method of characteristics (intro)
- Classification of second-order linear PDEs: elliptic, parabolic, hyperbolic
- Physical interpretation: equilibrium, diffusion, and wave propagation
- Well-posedness and types of boundary conditions (Dirichlet, Neumann)
Separation of variables
- Separation of variables for linear PDEs with homogeneous BCs
- Eigenvalue problems arising from separation
- Superposition and the general solution structure
- Sturm–Liouville theory (introduction at undergraduate level)
- Laplace's equation in a rectangle; Poisson equation (brief)
Classical PDEs in one space dimension
- Heat equation on a finite rod; Fourier sine/cosine series solutions
- Wave equation on a finite interval; d'Alembert's formula (1D)
- Vibrating string and heat conduction interpretations
- Steady-state solutions and equilibrium limits
- Inhomogeneous problems via eigenfunction expansion (introduction)
STEM / applied
Applied boundary-value problems
- Heat conduction in a slab; insulated and fixed-end boundaries
- Wave propagation: reflections, standing waves, and resonance
- Laplace's equation in simple domains; electrostatics analogy
- Method of images or maximum principle (introductory applications)
- Numerical visualization of solutions with software
Engineering interpretation
- Deriving the heat and wave equations from physical principles (overview)
- Boundary conditions from physical constraints (fixed, free, insulated)
- Energy methods and physical meaning of eigenmodes
- Comparison of parabolic vs hyperbolic behavior in applications
- Brief introduction to numerical PDE methods (finite differences)
Notes
Topics reflect common introductory PDE syllabi at US colleges and universities, emphasizing classical methods rather than graduate functional analysis. Exact coverage of first-order methods and 2D problems varies by program.