### Class Diary for M435, Spring 2007, Jochen Denzler

Wed Jan 10: Intro: what is a PDE? Notions of order, linear (hom/inhom), initial and bdry conditions; trivial examples (cf Farlow lesson 1). Hwk: Review from MV integral calculus the divergence theorem, b/c we'll need it soon, and crucially
Fri Jan 12: Derivation of the heat equation; initial boundary value problems; 3 kinds of BCs. Laplace eqn as steady state case of heat eqn. (course notes pp1-3; also ZT Ch6, sec1-3)
Mon Jan 15: MLK DAY
Wed Jan 17: The Laplace operator in polar coordinates and other curvilinear coordinates (course notes pp 4-6) Hwk 1-6 due Mon
Fri Jan 19: Laplace in curvilinear cordinates finished up; spherical coordinates. Some comments on Hwk 4.
Mon Jan 22: More comments on Hwk 1-6 deadline extended: now due Wed. Review of Euler's formula. Harmonic functions; (ZT VII.2) In particular harmonic functions depending only on the distance from some origin. Hwk: Read ZT VII.2
Wed Jan 24: Separation of variables: hypotheses for applicability. Practcal: cartesian and polar coordinates
Fri Jan 26: Separation of variables: periodicity condition for angle and discrete separation constant; Euler's equation for the radial coordinate. Separation in spherical coordinates (not all ODEs solved here). Hwk: 7 and 9 due Mon, if possible, else Wed
Mon Jan 29: BVP for Laplace eqn: calc in disk (and facts about general domains) Hwk: 8,10,12 due Wed or Fri
Wed Jan 31: Introduction to Fourier series for 2pi periodic functions: The claim; how the coeffs must look; and the `basis interpretation' via inner product on a vector space of functions.
Fri Feb 02: Poisson's formula derived from (tentative) Fourier series approach. Hwk: 15, 16, 17 due Wed
Mon Feb 05: More discussion (Q&A) of Poisson formula and Hwk 15-17. Proof of validity of Poisson formula begun. Concentration phenomenon as an obstruction to interchange limit and integral. Hwk: 13,14 due Fri
Wed Feb 07: Validity of Poisson formula proved. Informal introduction to Dirac delta and view of bdry data as superposition of `pulses'. Uniqueness of soln to BVP via maximum principle outlined.
Fri Feb 09: Proof of uniqueness via maximum principle (details worked out). Discussion of past and pending hwk (extended till Mon). Hints for New Hwk: 11 due Wed
Mon Feb 12: Review separation of variables. Hwk: 18-21 due next Mon. Smoothness of fct and decay of Fourier coefficients.
Wed Feb 14: The success of the Poisson formula proves convergence of Fourier series to the function for smooth (C^2) functions. Other results about Fourier series without proof; stated and explained.
Fri Feb 16: Study of Gibbs (overshoot) phenomenon in examples. Fourier series on intervals of arbitrary length. Extension from half-period interval to even or odd functions: cosine vs sine Fourier series.
Mon Feb 19: BCs in separation of variables determine which kind of Fourier expansion use. All homogeneous data are built in the separation of variables framework. Begun pg 191 of ZT
Wed Feb 21: Representation formula and mean value property for harmonic functions.
Fri Feb 23: Comments on Hwk 21. Outline how to use representation formula to solve all Dirichlet BVP's for a domain, provided a certain family of BVP's (in space: with bdry data 1/|x-x'|) can be solved.
Mon Feb 26: Green's function begun. Hwk 18-21 hard deadline Wed
Wed Feb 28: Green's function in disc
Fri Mar 02: EXAM 1
Mon Mar 05: Green's function in disk and ball. Hwk 24,26 asap. Next 22,23,25
Wed Mar 07: Some final wisdom about Green's function; heat equation started.
Fri Mar 09: Exam back. Separation of variables for HE.
Mon Mar 12: SPRING BREAK
Wed Mar 14: SPRING BREAK
Fri Mar 16: SPRING BREAK
Mon Mar 19: Separation of variables for HE da capo. Consequences: immediate smoothness of solutions. No general solvability backwards in time.
Wed Mar 21: Heat equation with inhomogeneous (time-independent) boundary conditions. The heat equation on a finite (or infinite) interval with a delta(x) initial condition (started).
Fri Mar 23: The heat kernel calculated.
Mon Mar 26: (Non-)uniqueness example for IVP of HE on real line (sketch); modeling questions; Uniqueness for bounded solutions to HE on the real line, and for solutions on a finite interval with Dirichlet BC.
Wed Mar 28: The Fourier transform. Hwk: Deadline for all pending hwk (22-26) is Monday
Fri Mar 30: Multi-dim. heat equation, eigenvalue problem for the Laplace operator.
Mon Apr 02: Exam back; EVP for Laplace in rectangle discussed. Have a look at Hwk 27-29 and ask questions Wed. Deadline some time soon after Easter
Wed Apr 04: Vibrations of a membrane or string; the wave equation and separation of variables for it.
Fri Apr 06: GOOD FRIDAY
Mon Apr 09: Some comments on pending hwk. Light cone; solutions f(x-ct), f(x+ct) of 1dim wave eqn. Initial value problem.
Wed Apr 11: Examples for 1dim wave eqn solution formula; IVP for half-line; 3dim wave eqn in radially symmetric case.
Fri Apr 13: more on radially symmetric 3im wave equation; intro to the idea/method of shperical means. (Physical principles behind it)
Mon Apr 16: Kirchhoff's formula obtained.
Wed Apr 18: Kirchhoff's formula finished up. Discussion of pending hwk.
Fri Apr 20: Quick review for exam; soln Hwk 29. Evaluation
Mon Apr 23: EXAM 2: max 2 hours within the 4pm-8pm interval; class at regular time cancelled
Wed Apr 25: Descent for wave eqn from 3D to 2D
Fri Apr 27: descent finished up; exam back.
Mon Apr 30: STUDY PERIOD
Wed May 02: FINAL EXAM: 2:45-4:45 scheduled by university policy