Advanced differential equations
Graduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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$1,162 · Advanced differential equations · 18 tutoring hrs
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Topics typically covered
Standard syllabus
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
STEM / applied
Applied PDE and numerical coupling
- Reaction–diffusion models and traveling waves
- Wave propagation and hyperbolic systems (linear, introduction)
- Numerical methods for PDEs: stability and consistency
- Inverse problems for PDEs (introduction)
- Multiscale models coupling ODEs and PDEs
Domain applications
- Fluid mechanics: Navier–Stokes at the mathematical level (introduction)
- Elasticity and vibration of membranes/rods
- Heat conduction and phase change models
- Population models with diffusion
- Research problem sets aligned to science and engineering graduate courses
Notes
Topics reflect common graduate differential equations syllabi at US universities. Functional-analytic depth varies; this list targets a standard first-year graduate DE sequence without full graduate PDE theory.