Standard syllabus
Partial differential equations · Undergraduate · Math
Topics
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)
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$1,162 · Partial differential equations · 18 tutoring hrs
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