Math611 - 600: Introduction to Ordinary and Partial Differential Equations, Fall 2020

This is the semesterlog for the course. It will contain information such as material covered, homework assignments, exams, etc. Course information and syllabus are here.

Semester log

Wednesday, August 19: Review of the derivative for maps Rⁿ to R^m. HW: problems 1,2 on page 13 of Classnotes1.
The text the department uses for Math409/410 is Thomson/Bruckner/Bruckner, Elementary Real Analysis; it contains much of the requiste background and is useful as a reference (integration in Rⁿ is missing). You can download the hyperlinked version for free here.

The first part of the course, on ordinary differential equations, loosely follows sections 1,2 4-7 of Michael Taylor, Partial Differential Equations (Basic Theory). This text is available for free download (from an A&M account) here. Taylor is very concise. Two other texts are Ordinary Differential Equations, by V. Arnold (MIT Press), and Ordinary Differential Equations, by Wolfgang Walter, Springer GTM. The site has a link for free download.
Monday, August 24: Finish derivative; C([a,b]) as a metric space.
Wednesday, August 26: Finish C([a,b]) as a metric space (this includes the contraction mapping principle); begin systems of ODEs, fundamental local existence and uniqueness theorem; HW: problems 1,2,3 on pages 1j-1k of Classnotes3.
Monday, August 31: Proof of local existence and uniqueness theorem, global uniqueness; single first order inhomogeneous equation; single higher order equation as a linear system; HW is problems 1, 2 on pages 4-4a of Classnotes4.
Wednesday, September 2: Exponentiation of matrices; inhomogeneous linear systems with constant coefficients; HW is problems 1-4 on pages 12-13 of Classnotes5.
Monday, September 7: finish inhomogeneous linear systems with constant coefficients; exponentiation of diagonalizable matrices, implications for solving constant coefficient linear systems; HW is problems 1-3 on pages 10-11 of Classnotes6.
Wednesday, September 9: solution of constant coefficient systems via upper traingular form; variable coefficient linear systems; HW is problems 2, 3 on pages 13-14 in Classnotes7 (due Monday).
Monday, September 14: Duhamel's principle; the set of solutions to the homogeneous system as a vector space, fundamental systems, the Wronski determinant; HW is problems 1,3 on pages 13-14 of Classnotes7, problem 3 on page 5 of Classnotes8.
Wednesday, September 16: finish Wronski determinant; begin dependence of solutions on initial conditions and parameters; HW is to catch up on previous assignments in preparation for Exam 1 (to be given out Wednesday, September 23, will be due Monday, September 28).
Monday, September 21: continue dependence on initial conditions and parameters, joint differentiability; HW: Write down the details in the last step in the proof of Proposition 6.1, read remainder (from class) of section 6.
Wednesday, September 23: flows and vector fields, straightening of vector fields. Give out Exam 1, due Monday, September 28.
Here are two very nice texts on ODEs: Ordinary Differential Equations, by V. Arnold (MIT Press), and Ordinary Differential Equations, by Wolfgang Walter, Springer GTM. The site has a link for free download.
This concludes the ODE portion of the class.
Monday, September 28: Exam 1 due today by the beginning of class.

We now start the PDE portion of the class, Evans chapters 1,2, and 3.1, 3.2.
Today: chapter 1 (this is a very brief introduction, with much of it meant for individual reading); 2.1, the transport equation; begin 2.2.1, Laplace's equation. HW: pages 12-13 (Evans), problems 4,5, page 85, problem 1.
Wednesday, September 30: 2.2.1 Laplace's equation, fundamental solutions, solving Poisson's equation; HW is problem 2, page 85.
Monday, October 5: class cancelled, make up time TBA.
Wednesday, October 7: 2.2.2 mean value formulas; 2.2.3 properties of harmonic functions (maximum principle, uniqueness in Dirichlet problem); HW problem 3, page 85 plus two exercises assigned in class.
Monday, October 12: continue properties of harmonic functions (smoothness, estimates on derivatives, Liouville's theorem, a representation formula); HW is problems 4,5 on page 85.
Wednesday, October 14: Harnack's inequality for harmonic functions, Green's function; HW as assigned in class notes.
Here are a couple of additional texts that I like. Introduction to partial differential equations, by Gerald Folland, and Basic linear partial differential equations, by Francois Treves (this is an inexpensive Dover edition).
Monday, October 19: Green's function and Poisson kernel for the unit ball; HW is problem 11, page 87. Exam 2 announced for Wednesday, October 28 (due Monday, November 2).
Wednesday, October 21: Solution of the Dirichlet problem on the ball, Poisson kernel as an approximation to the identity; HW is problem 2 on page 5a of the classnotes (17); start section on the heat equation, fundamental solution.
Monday, October 26: finish fundamental solution, the heat kernel as an approximation to the identity; solution of homogeneous equation with general initial condition, solution of inhomogeneous equation with homogeneous initial condition; HW provide some details in the proof of Theorem 2, as assigned in class.
Wednesday, October 28: finish inhomogeneous equation, solution of inhomogeneous problem with general initial data; begin mean value formula, heat balls; Exam 2 given out, due Monday, November 2.
Monday, November 2: finish mean value formula for heat equation; begin chapter 3, characteristics; Exam 2 due today. HW is problem 13, page 87.
Wednesday, November 4: derive characteristic equations, structure of characteristis ODEs, a fully nonlinear example; flattening the boundary; HW is problem 4 on page 162 plus the problem on page 7 of the lecture notes.
Monday, November 9: non-characteristic triples/initial conditions, "solving" the characteristic equations, begin proof of local existence theorem; HW is problem 7, page 163.
Wednesday, November 11: finish proof of local existence theorem; discuss Exam 1; HW is problem 5 on page 162, also: solve problem 2 on Exam 2 using the method of characteristics, compare to solution suggested in the hint on the exam).
Monday, November 16: some examples; begin 3.1, complete integrals; HW is problem 1, page 161.
Wednesday, November 18: new solutions via envelopes.
Monday, November 23: (last day of fall semester classes), ake home Exam 3, no class.