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

Vanderbilt












Marcelo Mendes Disconzi
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.  

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, 2:203:35pm 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.

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.
 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.  


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