TMA4305 Partielle Differensialligninger 2014
TMA4305 Partial Differential Equations Fall 2014
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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 | |
|---|---|---|---|
| 34 | 4.5.1, 4.5.2, 3.1, 3.2, ? | The beginning of Ch.4 later. Different kinds of equations and problems. Laplace's Eq | |
| 35 | Ch 3. 109–118, 124–133 | Laplace's Eq | |
| 36 | Ch.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 | |
| 37 | From Ch. 2 | The 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. |
| 38 | Ch. 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. 5 | Duhamel (again!). Method of Stationary Phase. Equations of 2nd order | Classification, Characterisitics |
| 42 | Ch. 5 | Characteristic coordinates. Eqns of 2nd order. Symmetric matrices | |
| 43 | ch. 5 | Elliptic Equations again A Proof | Weak Solutions and Shocks |
| 44 | Ch. 4 | ||
| 45 | Ch. 4 | Breaking time with smooth data. Why the Atom Does not Collapse. Comments on Existence | Perron's Method |
| 46 | Repetiton with examples | Elliptic eqs. | |
| 47 | Repetition with Examples | Planned 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 |
|---|---|---|
| 35 | Exercise 1 | 3/2-1/2(xx+yy) misprinted |
| 36 | 3.1, 3.2. 3.3, 3.5 | Example 3.5 Example 3.3 |
| 37 | 3.8, 3.11, 2.2 and these | ?Is 2.2 misprinted? |
| 38 | 2.1, 2.8, 2.12, 2.14, 2.15 | 2.12 was lectured, done.Soln 2.14 |
| 39 | 2.13, 2.16, 2.19 | Ex. 2.19 comment |
| 40 | 3.21, 5.5a, 5.4, Svag lo"sn | Ex. 5.4 |
| 41 | 5.3, 5.16, 5.17 | Ex.: verify that the solution in Kirchhoff's formula takes the initial data |
| 42 | Exercises | 5.17 focus14.pdf. Ex.3: bx-2ay,x. Ex.2: invariant |
| 43 | 5.8, 5.9, 5.10, 5.12, Example | confusing: non-characteristic initial values means that these be given on a characteristic curve !? In 5.9 the other coordinate can be \eta = x |
| 44 | 4.1, 4.2, Examples | |
| 45 | 4.3, 4.6, 4.9, 4.13 | Ex. 4.6Answer |
| 46 | Exercises | Solution with vanishing viscosity. The question in 2.16 confusing: u = y/2. (The obtained formula cannot be ext.) |
| 47 | Last 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
NOTES, LINKS
- Dirichlet's Principle Notes
- The temperature of the earth earth.pdf
- Perron's Method perron.pdf
- Poisson's formula from Cauchy'scauchy.pdf
Referansegruppe
| Håkon Bølviken | haakosb [at] stud [dot] ntnu [dot] no |
| Emil Hansen | emilhan [at] stud [dot] ntnu [dot] no |
| Stine Vennemo | vennemo [at] stud [dot] ntnu [dot] no |