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

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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*

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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)

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Mon Jan 15: ** MLK DAY

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Wed Jan 17: ** The Laplace operator in polar coordinates and other
curvilinear coordinates (course notes pp 4-6) *Hwk 1-6 due Mon*

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Fri Jan 19: ** Laplace in curvilinear cordinates finished up; spherical
coordinates. Some comments on Hwk 4.

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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*

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Wed Jan 24: ** Separation of variables: hypotheses for applicability.
Practcal: cartesian and polar coordinates

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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*

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Mon Jan 29: ** BVP for Laplace eqn: calc in disk (and facts about general
domains) *Hwk: 8,10,12 due Wed or Fri*

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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.

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Fri Feb 02: ** Poisson's formula derived from (tentative) Fourier series
approach. *Hwk: 15, 16, 17 due Wed*

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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*

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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.

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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*

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Mon Feb 12: ** Review separation of variables.
*Hwk: 18-21 due next Mon*. Smoothness of fct and decay of Fourier
coefficients.

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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.

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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.

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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

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Wed Feb 21: ** Representation formula and mean value property for
harmonic functions.

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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.

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Mon Feb 26: ** Green's function begun. *Hwk 18-21 hard deadline Wed*

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Wed Feb 28: ** Green's function in disc

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Fri Mar 02: ** EXAM 1

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Mon Mar 05: ** Green's function in disk and ball.
*Hwk 24,26 asap. Next 22,23,25*

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Wed Mar 07: ** Some final wisdom about Green's function; heat equation
started.

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Fri Mar 09: ** Exam back. Separation of variables for HE.

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Mon Mar 12: ** SPRING BREAK

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Wed Mar 14: ** SPRING BREAK

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Fri Mar 16: ** SPRING BREAK

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Mon Mar 19: ** Separation of variables for HE da capo. Consequences:
immediate smoothness of solutions. No general solvability backwards
in time.

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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).

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Fri Mar 23: ** The heat kernel calculated.

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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.

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Wed Mar 28: ** The Fourier transform. *Hwk: Deadline for all pending
hwk (22-26) is Monday*

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Fri Mar 30: ** Multi-dim. heat equation, eigenvalue problem for the
Laplace operator.

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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*

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Wed Apr 04: ** Vibrations of a membrane or string; the wave equation
and separation of variables for it.

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Fri Apr 06: ** GOOD FRIDAY

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Mon Apr 09: ** Some comments on pending hwk.
Light cone; solutions f(x-ct), f(x+ct) of 1dim wave eqn.
Initial value problem.

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Wed Apr 11: ** Examples for 1dim wave eqn solution formula;
IVP for half-line; 3dim wave eqn in radially symmetric case.

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Fri Apr 13: ** more on radially symmetric 3im wave equation; intro
to the idea/method of shperical means. (Physical principles behind it)

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Mon Apr 16: ** Kirchhoff's formula obtained.

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Wed Apr 18: ** Kirchhoff's formula finished up. Discussion of pending hwk.

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Fri Apr 20: ** Quick review for exam; soln Hwk 29. Evaluation

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Mon Apr 23: ** EXAM 2: max 2 hours within the 4pm-8pm interval; class
at regular time cancelled

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Wed Apr 25: ** Descent for wave eqn from 3D to 2D

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Fri Apr 27: ** descent finished up; exam back.

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Mon Apr 30: ** STUDY PERIOD

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Wed May 02: ** FINAL EXAM: 2:45-4:45
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scheduled by university policy *

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