TMA4305 Partielle Differensialligninger 2014

TMA4305 Partial Differential Equations Fall 2014

Messages

Office hours before the exam:

Wednesday the 26th from 14 to 16 in room 1152, SB II
Thursday the 27th from 12 to 14 in room 1152, SB II

See below about the Exam.

* The lectures are in K26. (Kjemi 4, first floor)
About Ch. 2: Random walk almost omitted for Heat Eqn.

Week Section Comments
344.5.1, 4.5.2, 3.1, 3.2, ?The beginning of Ch.4 later. Different kinds of equations and problems. Laplace's Eq
35Ch 3. 109–118, 124–133Laplace's Eq
36Ch.3. 132–133, 139–155
Chapter 2.1,2.2
Thm 3.11 with proof. Lemma 3.2 by an example.Easy proof of Parabolic Maximum Principle Simplest case
37From Ch. 2The Heat Equation, Maximum principles, Heat kernel via Fourier Transf (Ex. 2.12, p.99). Initial values attained (rigorous proof). Duhamels Principle with verification. Einstein 1905 Fouriertransf.
38Ch. 2. Ch. 5 (?)Heat Eqn. Tychonov's Thm. Backwards uniqueness. Porous Medium Eqn.
39 Ch. 5 Wave Eqn., Energy, D'Alembert, Duhamel's Principle,Weak Solutions
40 Ch. 5 Kirchhoffs formula. Huygen's Principle
41 Ch. 5Duhamel (again!). Method of Stationary Phase. Equations of 2nd orderClassification, Characterisitics
42 Ch. 5Characteristic coordinates. Eqns of 2nd order. Symmetric matrices
43 ch. 5Elliptic Equations again A ProofWeak Solutions and Shocks
44Ch. 4
45Ch. 4Breaking time with smooth data. Why the Atom Does not Collapse. Comments on Existence Perron's Method
46Repetiton with examplesElliptic eqs.
47Repetition with ExamplesPlanned 2006/3,4. 2007/1,3,4. 2010/2. 2011/1. 2009/2,4.

Information about the course

Textbook

Sandro Salsa: "Partial Differential Equations in Action", Springer.

Misprints in the book

errata_salsa.pdf —Abundant dulcibus vitiis !

Habent sua fata libelli pro captu lectoris.

Lectures

Tuesday 12–14 in K26
Thursday 12–14 in K26

Teacher

Peter Lindqvist (SB II room 1152)

Exercises

*Monday 13 –14 S 265

Week Problems Comments
35Exercise 13/2-1/2(xx+yy) misprinted
363.1, 3.2. 3.3, 3.5 Example 3.5 Example 3.3
373.8, 3.11, 2.2 and these?Is 2.2 misprinted?
382.1, 2.8, 2.12, 2.14, 2.152.12 was lectured, done.Soln 2.14
392.13, 2.16, 2.19 Ex. 2.19 comment
403.21, 5.5a, 5.4, Svag lo"snEx. 5.4
415.3, 5.16, 5.17Ex.: verify that the solution in Kirchhoff's formula takes the initial data
42Exercises5.17 focus14.pdf. Ex.3: bx-2ay,x. Ex.2: invariant
435.8, 5.9, 5.10, 5.12, Exampleconfusing: non-characteristic initial values means that these be given on a characteristic curve !? In 5.9 the other coordinate can be \eta = x
444.1, 4.2, Examples
454.3, 4.6, 4.9, 4.13Ex. 4.6Answer
46 ExercisesSolution with vanishing viscosity. The question in 2.16 confusing: u = y/2. (The obtained formula cannot be ext.)
47Last exercises

Syllabus (Pensum)

ALL THE EXERCISES (Øvinger)
Chapter 2

2.1, 2.2, 2.3, not 2.4, not 2.5.1, 2.5.2, not 2.5.3, not 2.6, 2.8, not 2.9, 2.10.1, not 2.10.2.

Chapter 3

3.1, 3.2, not 3.3.1, 3.3.2 - 3.3.5. Not 3.36, not 3.3.7. 3.4, 3.5.1, not 3.5.2, 3.5.3, not 3.5.4, 3.6, 3.7.1 (Only Gauss's Lemma), not 3.7.2.

Chapter 4

4.1 - 4.5 (not 4.4.1, not 4.5.4), not 4.6.

Chapter 5

Not 5.6, not 5.8.2, not 5.9.1, not 5.10, but 5.10.6 (Stationary Phase) in the simplest case included.
Sections 5.1 and 5.4.5 serve as orientation.

How to find the Euler-Lagrange Equation of a Variational Integral. Dirichlet's Principle.

The End


Extra literature

P. Olver: "Introduction to Partial Differential Equations. Springer. –Available on Springer link. Interesting. L. C. Evans:"Partial Differential Equations". –One of the most used standard texts today. Advanced.
W. Strauss: "Partial Differential Equations: an Introduction". –Clear and easy to read.
A. Tveito & R. Winther:" Introduction to Partial Differential Equations (A Computational Approach)". –An excellent, but very elementary account.

Exam 1.XII.2014

The exam (written) is in English only. The students may answer in Norwegian or English. To the exam you are allowed to bring one A4-sized sheet of yellow paper on which you may write whatever you want in advance. It must be pre-stamped at the Department of Mathematics (SB II, 7th floor), where empty sheets can be acquired. No other aids are permitted in the exam.

Exam 1.XII.2014.Misprint 3b. Read u(x,y,z,0) = f(x,y,z). Solutions 1.XII.2014

Referansegruppe

2015-06-22, Hallvard Norheim Bø