Standard syllabus
Partial differential equations · Undergraduate · Math
Learning objectives from the Partial 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.
Undergraduate Partial Differential Equations — scope drawn from typical US intro PDE syllabi and open PDE notes (separation of variables, Fourier series, classic equations).
First-order equations and classification
- Partial derivatives and notation — Partial derivatives and notation; initial and boundary value problems
- First-order linear — First-order linear
- Quasilinear PDEs — Quasilinear PDEs
- Classification of second-order linear PDEs: elliptic, parabolic, hyperbolic — Classification of second-order linear PDEs: elliptic, parabolic, hyperbolic
- Physical interpretation — Physical interpretation
- Wave propagation — Wave propagation
- Well-posedness — Well-posedness
- Types of boundary conditions (Dirichlet, Neumann) — Types of boundary conditions (Dirichlet, Neumann)
Separation of variables
- Separation of variables for linear PDEs with homogeneous BCs — Separation of variables for linear PDEs with homogeneous BCs
- Eigenvalue problems arising from separation — Eigenvalue problems arising from separation
- Superposition — Superposition
- The general solution structure — The general solution structure
- Sturm–Liouville theory (introduction at undergraduate level) — Sturm–Liouville theory (introduction at undergraduate level)
- Laplace's equation in a rectangle — Laplace's equation in a rectangle; Poisson equation (brief)
Classical PDEs in one space dimension
- Heat equation on a finite rod — Heat equation on a finite rod; Fourier sine/cosine series solutions
- Wave equation on a finite interval — Wave equation on a finite interval; d'Alembert's formula (1D)
- Vibrating string — Vibrating string
- Heat conduction interpretations — Heat conduction interpretations
- Steady-state solutions — Steady-state solutions
- Equilibrium limits — Equilibrium limits
- Inhomogeneous problems via eigenfunction expansion (introduction) — Inhomogeneous problems via eigenfunction expansion (introduction)
Learning objectives
Click an objective for study materials.
First-order equations and classification
- Partial derivatives and notation — Partial derivatives and notation; initial and boundary value problems
- First-order linear — First-order linear
- Quasilinear PDEs — Quasilinear PDEs
- Classification of second-order linear PDEs: elliptic, parabolic, hype... — Classification of second-order linear PDEs: elliptic, parabolic, hyperbolic
- Physical interpretation — Physical interpretation
- Wave propagation — Wave propagation
- Well-posedness — Well-posedness
- Types of boundary conditions (Dirichlet, Neumann) — Types of boundary conditions (Dirichlet, Neumann)
Separation of variables
- Separation of variables for linear PDEs with homogeneous BCs — Separation of variables for linear PDEs with homogeneous BCs
- Eigenvalue problems arising from separation — Eigenvalue problems arising from separation
- Superposition — Superposition
- The general solution structure — The general solution structure
- Sturm–Liouville theory (introduction at undergraduate level) — Sturm–Liouville theory (introduction at undergraduate level)
- Laplace's equation in a rectangle — Laplace's equation in a rectangle; Poisson equation (brief)
Classical PDEs in one space dimension
- Heat equation on a finite rod — Heat equation on a finite rod; Fourier sine/cosine series solutions
- Wave equation on a finite interval — Wave equation on a finite interval; d'Alembert's formula (1D)
- Vibrating string — Vibrating string
- Heat conduction interpretations — Heat conduction interpretations
- Steady-state solutions — Steady-state solutions
- Equilibrium limits — Equilibrium limits
- Inhomogeneous problems via eigenfunction expansion (introduction) — Inhomogeneous problems via eigenfunction expansion (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.
- First-order equations and classification
Partial derivatives and notation; initial and boundary value problems
Coming soon - Separation of variables
Separation of variables for linear PDEs with homogeneous BCs
Coming soon - Classical PDEs in one space dimension
Heat equation on a finite rod; Fourier sine/cosine series solutions
Coming soon
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