This syllabus is entirely tentative, and is subject to change: this
version contains changes from its predecessor (Ch. 3 is out and Ch. 7
is in). It may undergo further modification as the semester progresses,
but a current version will always be available from the Section Home Page. Students who browse this
location should be aware that **this syllabus is
not 100% determinative **as currently posted.

This
preliminary version of the syllabus contains no selections of textbook
problems. ** Once the course
begins, there will be a list of problems that will be
adjusted throughout the semester.** *Students are responsible
for the content of all assigned problems; this content may occur in
examination questions.*

**Links to Additional notes** may
appear on the section home page. Some of these notes may contain
homework problems; these will be indicated in the syllabus.

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Chapter/Section | Topics |

Chapter 1 |
Where PDEs come from |

1.1 | What is a Partial Differential Equation? |

1.2 | First-order Linear Equations (Solution in the constant-coefficient case; the variable-coefficient case and characteristic curves. |

1.3 | Flows, Vibrations and Diffusions (Derivations of PDEs in various physical
situations; e.g., the vibrating string, the vibrating drumhead,
diffusion, heat flow, hydrogen atom). |

1.4 | Initial and boundary conditions (the Dirichlet, Neumann and Robin conditions and their significance for the vibrating string and diffusion equations. Conditions at infinity.) |

1.5 | Well- (and ill-)Posed Problems. |

1.6 | Types of second-order equations. |

Chapter 2 |
Waves and Diffusion |

2.1 | The Wave Equation (D'Alembert's solution on the line; the plucked string). |

2.2 | Causality and Energy. |

2.3 | The Diffusion (or Heat) Equation (the maximum principle; uniqueness for the Dirichlet problem). |

2.4 | Diffusion on the whole real line (the Gaussian or fundamental solution). |

2.5 | Comparison of waves and diffusion. |

Chapter 4 |
Boundary Problems |

4.1 | Separation of Variables, the Dirichlet Condition (both for the wave and the diffusion equations). |

- | Read §§ 5.1, 5.2 and 5.3 now! §5.1 problems are assigned. |

4.2 | The Neumann Condition. |

4.3 | Robin's Conditions (cases in which zero is an eigenvalue and cases in which one eigenvalue is negative). |

Chapter 5 |
Fourier Series |

5.1 | The Coefficients (or discrete Fourier transform): formulas for the coefficients, applications to the wave and the diffusion equations. |

5.2 | Even, Odd, Periodic and Complex-valued functions. |

5.3 | Orthogonality and "General Fourier Series" (orthogonal systems from symmetric boundary conditions; complex eigenvalues), |

5.4 | Completeness (three notions of convergence: pointwise, uniform and mean-square: convergence results for Fourier series and their generalizations). |

5.5 | Completeness and the Gibbs phenomenon. |

5.6 | Inhomogeneous Boundary Conditions. |

Chapter 6 |
Harmonic Functions |

6.1 | The Laplace Equation (its physical significance, maximum principle, uniqueness of solutions of the Dirichlet Problem, invariance of the Laplace operator under rigid motions). |

6.2 | Rectangles and Cubes. |

Chapter 7 |
Green's Identities and Green's Functions |

7.1 | Green's First Identity (and some consequences). |

7.2 | Green's Second Identity (and some consequences). |

7.3 | Green's Functions and the Dirichlet Problem. |

7.4 | Half-Spaces and Spheres. |

last revised 1245 EST 02/15/2001