Standard syllabus
Advanced differential equations · Graduate · Math
Learning objectives from the Advanced differential equations syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Advanced Differential Equations — scope drawn from open advanced DE / dynamical systems notes and typical US graduate advanced DE syllabi.
ODE theory
- Existence and uniqueness: Picard–Lindelöf and Peano theorems — Existence and uniqueness: Picard–Lindelöf and Peano theorems
- Continuation of solutions — Continuation of solutions
- Maximal intervals — Maximal intervals
- Linear systems: matrix exponential and Jordan form — Linear systems: matrix exponential and Jordan form
- Stability via linearization — Stability via linearization
- Lyapunov methods — Lyapunov methods
- Perturbation of ODEs — Perturbation of ODEs
- Averaging (introduction) — Averaging (introduction)
PDE foundations
- Weak derivatives — Weak derivatives
- Sobolev spaces (introduction) — Sobolev spaces (introduction)
- Classical, weak, and strong solutions — Classical, weak, and strong solutions
- Maximum principles for elliptic — Maximum principles for elliptic
- Parabolic equations — Parabolic equations
- Energy methods and uniqueness — Energy methods and uniqueness
- Green's functions — Green's functions
- Fundamental solutions — Fundamental solutions
Spectral and transform methods
- Sturm–Liouville theory — Sturm–Liouville theory
- Eigenfunction expansions — Eigenfunction expansions
- Separation of variables for canonical PDEs — Separation of variables for canonical PDEs
- Fourier and Laplace transform methods for PDEs — Fourier and Laplace transform methods for PDEs
- Distribution theory for PDEs (introduction) — Distribution theory for PDEs (introduction)
- Well-posedness for initial-boundary value problems — Well-posedness for initial-boundary value problems
Applied PDE and numerical coupling
- Reaction–diffusion models — Reaction–diffusion models
- Traveling waves — Traveling waves
- Wave propagation and hyperbolic systems (linear, introduction) — Wave propagation and hyperbolic systems (linear, introduction)
- Numerical methods for PDEs — Numerical methods for PDEs
- Consistency — Consistency
- Inverse problems for PDEs (introduction) — Inverse problems for PDEs (introduction)
- Multiscale models coupling ODEs and PDEs — Multiscale models coupling ODEs and PDEs
Domain applications
- Fluid mechanics: Navier–Stokes at the mathematical level (introduction) — Fluid mechanics: Navier–Stokes at the mathematical level (introduction)
- Elasticity — Elasticity
- Vibration of membranes/rods — Vibration of membranes/rods
- Heat conduction and phase change models — Heat conduction and phase change models
- Population models with diffusion — Population models with diffusion
- Research problem sets aligned to science — Research problem sets aligned to science
- Engineering graduate courses — Engineering graduate courses
Learning objectives
Click an objective for study materials.
ODE theory
- Existence and uniqueness: Picard–Lindelöf and Peano theorems — Existence and uniqueness: Picard–Lindelöf and Peano theorems
- Continuation of solutions — Continuation of solutions
- Maximal intervals — Maximal intervals
- Linear systems: matrix exponential and Jordan form — Linear systems: matrix exponential and Jordan form
- Stability via linearization — Stability via linearization
- Lyapunov methods — Lyapunov methods
- Perturbation of ODEs — Perturbation of ODEs
- Averaging (introduction) — Averaging (introduction)
PDE foundations
- Weak derivatives — Weak derivatives
- Sobolev spaces (introduction) — Sobolev spaces (introduction)
- Classical, weak, and strong solutions — Classical, weak, and strong solutions
- Maximum principles for elliptic — Maximum principles for elliptic
- Parabolic equations — Parabolic equations
- Energy methods and uniqueness — Energy methods and uniqueness
- Green's functions — Green's functions
- Fundamental solutions — Fundamental solutions
Spectral and transform methods
- Sturm–Liouville theory — Sturm–Liouville theory
- Eigenfunction expansions — Eigenfunction expansions
- Separation of variables for canonical PDEs — Separation of variables for canonical PDEs
- Fourier and Laplace transform methods for PDEs — Fourier and Laplace transform methods for PDEs
- Distribution theory for PDEs (introduction) — Distribution theory for PDEs (introduction)
- Well-posedness for initial-boundary value problems — Well-posedness for initial-boundary value problems
Applied PDE and numerical coupling
- Reaction–diffusion models — Reaction–diffusion models
- Traveling waves — Traveling waves
- Wave propagation and hyperbolic systems (linear, introduction) — Wave propagation and hyperbolic systems (linear, introduction)
- Numerical methods for PDEs — Numerical methods for PDEs
- Consistency — Consistency
- Inverse problems for PDEs (introduction) — Inverse problems for PDEs (introduction)
- Multiscale models coupling ODEs and PDEs — Multiscale models coupling ODEs and PDEs
Domain applications
- Fluid mechanics: Navier–Stokes at the mathematical level (introduction) — Fluid mechanics: Navier–Stokes at the mathematical level (introduction)
- Elasticity — Elasticity
- Vibration of membranes/rods — Vibration of membranes/rods
- Heat conduction and phase change models — Heat conduction and phase change models
- Population models with diffusion — Population models with diffusion
- Research problem sets aligned to science — Research problem sets aligned to science
- Engineering graduate courses — Engineering graduate courses
Definitions and structure
- Existence and uniqueness: Picard–Lindelöf and Peano theorems — Existence and uniqueness: Picard–Lindelöf and Peano theorems
- Continuation of solutions — Continuation of solutions
- Maximal intervals — Maximal intervals
- Linear systems: matrix exponential and Jordan form — Linear systems: matrix exponential and Jordan form
- Stability via linearization — Stability via linearization
Proofs and reasoning
- Lyapunov methods — Lyapunov methods
- Perturbation of ODEs — Perturbation of ODEs
- Averaging (introduction) — Averaging (introduction)
- Weak derivatives — Weak derivatives
- Sobolev spaces (introduction) — Sobolev spaces (introduction)
Abstraction and generalization
- Classical, weak, and strong solutions — Classical, weak, and strong solutions
- Maximum principles for elliptic — Maximum principles for elliptic
- Parabolic equations — Parabolic equations
- Energy methods and uniqueness — Energy methods and uniqueness
- Green's functions — Green's functions
Modeling and computation
- Apply existence and uniqueness: picard–lindelöf and peano theorems in... — Apply existence and uniqueness: picard–lindelöf and peano theorems in engineering contexts
- Apply continuation of solutions in engineering contexts — Apply continuation of solutions in engineering contexts
- Apply maximal intervals in engineering contexts — Apply maximal intervals in engineering contexts
- Apply linear systems: matrix exponential and jordan form in engineeri... — Apply linear systems: matrix exponential and jordan form in engineering contexts
- Apply stability via linearization in engineering contexts — Apply stability via linearization in engineering contexts
Data and technology
- Use software to explore advanced differential equations problems nume... — Use software to explore advanced differential equations problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Lyapunov methods — Lyapunov methods
- Perturbation of ODEs — Perturbation of ODEs
- Averaging (introduction) — Averaging (introduction)
- Weak derivatives — Weak derivatives
- Sobolev spaces (introduction) — Sobolev spaces (introduction)
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- ODE theory
Existence and uniqueness: Picard–Lindelöf and Peano theorems
Coming soon - PDE foundations
Weak derivatives
Coming soon - Spectral and transform methods
Sturm–Liouville theory
Coming soon - Applied PDE and numerical coupling
Reaction–diffusion models
Coming soon - Domain applications
Fluid mechanics: Navier–Stokes at the mathematical level (introduction)
Coming soon
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$1,162 · Advanced differential equations · 18 tutoring hrs
Study guides, worksheets, reviews, practice tests, and answer keys for 1 class. 18 tutoring hours (1 hr / week · semester). Bundle discount applied vs buying separately. Pay in full via Zelle.