Standard syllabus
Advanced differential equations · Graduate · Math
Topics
ODE theory
- Existence and uniqueness: Picard–Lindelöf and Peano theorems
- Continuation of solutions and maximal intervals
- Linear systems: matrix exponential and Jordan form
- Stability via linearization and Lyapunov methods
- Perturbation of ODEs and averaging (introduction)
PDE foundations
- Weak derivatives and Sobolev spaces (introduction)
- Classical, weak, and strong solutions
- Maximum principles for elliptic and parabolic equations
- Energy methods and uniqueness
- Green's functions and fundamental solutions
Spectral and transform methods
- Sturm–Liouville theory and eigenfunction expansions
- Separation of variables for canonical PDEs
- Fourier and Laplace transform methods for PDEs
- Distribution theory for PDEs (introduction)
- Well-posedness for initial-boundary value problems
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$1,162 · Advanced differential equations · 18 tutoring hrs
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