Marcelo Mendes Disconzi

Department of Mathematics, Vanderbilt University

email: marcelo.disconzi at vanderbilt.edu

office: Stevenson Center 1515

phone: (615) 322 7147 fax: (615) 343 0215

mail to: 1326 Stevenson Center Ln, Vanderbilt University, Nashville TN 37240

Department of Mathematics, Vanderbilt University

email: marcelo.disconzi at vanderbilt.edu

office: Stevenson Center 1515

phone: (615) 322 7147 fax: (615) 343 0215

mail to: 1326 Stevenson Center Ln, Vanderbilt University, Nashville TN 37240

MATH 3120- Introduction of partial differential equations

General
Information

For a description of the course, including the grading policy, consult the course syllabus. Students are responsible for reading the syllabus and being aware of all the course and university policies. If you are taking this course for graduate credit, consult the MATH 5120 syllabus.

For a description of the course, including the grading policy, consult the course syllabus. Students are responsible for reading the syllabus and being aware of all the course and university policies. If you are taking this course for graduate credit, consult the MATH 5120 syllabus.

This
course will be taught through a hybrid online/in-person model. See the course syllabus for details.

Textbook: No textbook will be required. See the course syllabus for suggested references.

Classes meet on TR, 2:20–3:35pm at Buttrick Hall 206.

Due to
the COVID-19
outbreak, we will be following all guidelines stipulated by
Vanderbilt University. See the return
to campus website, the Vanderbilt
COVID-19 response website, and the Vanderbilt
COVID-19 dashboard for more information.

Contact
Information and Office Hours

Instructor's office: Stevenson Center 1515 (5th floor of the Mathematics Building).

Instructor's email: marcelo.disconzi@vanderbilt.edu.

Instructor's office hours: Tuesdays, 4-5pm, Thursdays, 1-2pm and 5-6pm, or by appointment. Office hours will be held virtually. If you want to meet in person, please schedule an appointment.

Instructor's office phone: (615) 322-7147.

Instructor's office: Stevenson Center 1515 (5th floor of the Mathematics Building).

Instructor's email: marcelo.disconzi@vanderbilt.edu.

Instructor's office hours: Tuesdays, 4-5pm, Thursdays, 1-2pm and 5-6pm, or by appointment. Office hours will be held virtually. If you want to meet in person, please schedule an appointment.

Instructor's office phone: (615) 322-7147.

Schedule

Below is an ongoing schedule for the course (for the academic calendar,
click here).
Click here for the class notes, and
here for the class notes in handwritten form.
The class notes will be updated and typeset as the course progress. Since typesetting
them might take some time, typically the most up-to-date version of the notes will be
in the handwritten form.

Date | Material covered | Remarks |
---|---|---|

Jan 26 | Introduction. Examples of PDEs. | HW 1 is posted on Brightspace. Due: Feb 2, 11:59pm. Solutions to HW 1. |

Jan 28 | Separation of variables for the Schrodinger equation with a radially symmetric potential. | |

Feb 2 | More on the Schrodinger equation. The angular equation (Legendre polynomials and spherical harmonics). The radial equation. | HW 2 is posted on Brightspace. Due: Feb 9, 4pm. Solutions to HW 2. |

Feb 4 | Final remarks on the Schrodinger equation. Separation of variables for the one-dimensional wave equation. | |

Feb 9 | More on the wave equation. Fourier series. | HW 3 is posted on Brightspace. Due: Feb 16, 4pm. Solutions to HW 3. |

Feb 11 | Convergence of Fourier series. | |

Feb 16 | Sine and cosine Fourier series. Back to the initial-value-boundary problem for the 1d wave equation. The 1d wave equation on the real line; D'Alembert's formula. | HW 4 is posted on Brightspace. Due: Feb 22, 11:59pm. Solutions to HW 4. |

Feb 18 | Domains of influence and dependence for the 1d wave equation. Generalized solutions to the 1d wave equation and propagation of singularities. Some general tools for the study of PDEs. | HW 5 is posted on Brightspace. Due: Mar 2, 4:00pm. Solutions to HW 5. Extra credit 1 is posted on Brightspace. Due: Mar 2, 4:00pm. |

Feb 23 | Formal aspects of PDEs (calculus facts, multi-index notation, some useful conventions, etc.) | In-class reading day. |

Feb 25 | The fundamental solution to the Laplacian in R^n. Solutions to Poisson's equation in R^n. | In-class reading day. |

Mar 2 | Harmonic functions. The mean value theorem and its converse. | HW 6 is posted on Brightspace. Due: Mar 9, 4:00pm. Solutions to HW 6. Extra credit 2 is posted on Brightspace. Due: Mar 11, 11:59pm. |

Mar 4 | The wave equation in R^n. Finite propagation speed. | |

Mar 9 | Toward solutions to the wave equation in R^n. Spherical averages and the Euler-Poisson-Darboux equation. Reflection method. | HW 7 is posted on Brightspace. Due: Mar 16, 4:00pm. Solutions to HW 7. |

Mar 11 | Existence and uniqueness of solutions to the Cauchy problem for the wave equation in n=2,3. Kirchhoff's and Poisson's formulas. | |

Mar 16 | Solutions to the inhomogeneous wave equation in R^n; Duhamel's principle. Vectorfields as differential operators. The Minkowski metric. | HW 8 is posted on Brightspace. Due: Mar 23, 4:00pm. Solutions to HW 8. HW 9 (Heat eq. project) is posted on Brightspace. Due: Mar 30, 4:00pm. Solutions to HW 9. |

Mar 18 | Lorentz vectorfields. Differential operators and commutators. Sobolev inequality. | |

Mar 23 | Decay estimates for solutions to the wave equation in R^n. | |

Mar 25 | The canonical form of second order linear PDEs: elliptic, parabolic, and hyperbolic equations. General strategy for the study of PDEs. Well-posedness. The method of characteristics. | HW 10 is posted on Brightspace. Due: Apr 6, 4:00pm. Solutions to HW 10. |

Mar 30 | Local existence and uniqueness for quasilinear first-order PDEs via the method of characteristics. | |

Apr 1 | Bugers' equation. Blow-up and shocks. | HW 11 is posted on Brightspace. Due: Apr 13, 4:00pm. Solutions to HW 11. |

Apr 6 | Scalar conservation laws in one dimension. Weak solutions. The Rankine-Hugoniot conditions. | The class projects are posted on Brightspace. Due: May 5, 4:00pm. |

Apr 8 | Systems of conservation laws in one dimension. Simple waves. Rarefaction waves. | In-class reading day. |

Apr 13 | The Riemann problem. Riemann invariants. | HW 12 is posted on Brightspace. Due: Apr 20, 4:00pm. Solutions to HW 12. |

Apr 15 | Breakdown of solutions to 2x2 systems of conservation laws. Non-uniqueness of weak solutions. Entropy solutions. | |

Apr 20 | Review/class project. | |

Apr 22 | Review/class project. | |

Apr 27 | Review/class project. | |

Apr 29 | Review/class project. |

Anonymous
feedback

Students are encouraged to bring suggestions and discuss with the course instructor any concerns they may have, including something they think is not being properly handled in the course. But if you do not feel comfortable doing that, here you have the opportunity to send anonymous feedback.

Students are encouraged to bring suggestions and discuss with the course instructor any concerns they may have, including something they think is not being properly handled in the course. But if you do not feel comfortable doing that, here you have the opportunity to send anonymous feedback.