School of Mathematics and Computing Sciences, Chalmers University of Technology and Goteborg University


TMA371/MAN660 Partial Differential Equations TM/GU, Fall 2000


General information:

This is the first course on partial differential equations in Engineering Mathematics program, Teknisk matematik, at Chalmers. Of course students who are not following this program are also welcome. The course consists of 35 lectur hourse, 14 exercise hourse and 21 hours of computer exercises and gives 4 points.


Teachers:

Mohammad Asadzadeh (Lectures/Exercises)
telephone: 7723517
e-mail:mohammad@math.chalmers.se
URL:http://www.math.chalmers.se/~mohammad
Matematiskt Centrum, Eklandagatan, room 2331

Tommy Gustafsson (Exercises)
telefon: 772 5306
e-mail:tommyg@math.chalmers.se
URL:http://www.math.chalmers.se/~tommyg
Matematiskt Centrum, Eklandagatan, room 2226

Niklas Ericsson (Computer assignments)
telephone: 772 53 19
e-mail:nen@math.chalmers.se
URL:http://www.math.chalmers.se/~nen
Matematiskt Centrum, Eklandagatan 80, ASECO


Where and when: Lectures: Tuesdays 13.15-17 and Thursdays 8.00-9.45
Exercises: Thursdays 10-11.45.

24/10, 13-17; in VB
26/10, 08-12; inVM
31/10, 13-17; in VB
02/11, 08-12; in VB
07/11, 13-17; in VB
09/11, 08-12; in HC2
14/11, 13-17; in VB
16/11, 08-12; in HC2; OBS! OBS! OBS! today, no lecture 8-10
21/11, 13-17; in VB
23/11, 08-12; in HC2
28/11, 13-17; in VF
30/11, 08-12; in VB
05/12, 13-17; in VB
07/12, 08-12; in VA

E3 Students have a separate program covering the first week schema:

Monday 23/10, Tuesday 24/10, Thursday 26/10, Monday 30/10

Time: 17.30-20.00; Place MD3

This particular schema for E3 students is because of their project work during the first week. It also covers solving partial differential and integral equations using Laplace and Fourier transforms that is, partially, removed from E2's Fourier analysis course.


Literature:

K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational Differential Equations, Studentliterature 1996 (available at the Cremona book shop).


Home Assignments (Voluntary):

During the course two sets of assignments will be handed out. Each set will be divided in two parts. One part contains problems of the same type as on the final exam, but more complex and time consuming, the other part consists of computer exercises. You will account for your work by handing in the solutions to the problems and a short report on the computer work done. The home assignments can give a maximum of 12 bonus points on the final exam, where the computer exercises give 0-4 points each and the problems give 0-2 points each.


$\circ$ Assignment 1a (handed in no later than Tuesday week5).

$\circ$ Assignment 1b (handed in no later than Tuesday week5).

$\circ$ Assignment 2a (handed in no later than Thursday week7).

$\circ$ Assignment 2b (handed in no later than Thursday week7).


The computer assignments:

The computer assignments use Matlab as the main tool. You can sit at MC if you want to (Niklas can give you a log-in name), but as long as you have access to Matlab 5.2 you can sit anywhere.

Apart from the two bonus generating assignments, we have also made some preparatory exercises, assignment 0. They deal with interpolation (ch 5 in the book) and multidimensional calculus (ch 13). Take a look at them to see if you can learn something from them, probably you will.

Assignment 0 (voluntary)
Assignment1b (bonus generating)
Assignment2b(bonus generating)


Old exams with solutions:

Here are two examples of old exams, (2000-04-25) , (99-04-06) and (98-12-15)
with their solutions: solution to (200-04-25) , solution to (99-04-06) and solution to (98-12-15) .

Examples of two other old exams, with solutions, will be appear on the web-site during the course.


Final Exam (Compulsary):

The final exam is written and will contain both problems and theory questions. The grades on the exam will be:



$\circ$ 3: 20-29 points

$\circ$ 4: 30-39 points

$\circ$ 5: 40-50 points



The exam will be December 13, 14.15-18.15 in VV, no aids allowed. The corrected exams will be available in the reception room at Matematiskt Centrum which is open during lunch time, Monday to Friday, no later than three weeks after the exam.


The final exam will contain a theory question from the following list:

Theorem 8.1; a priori error estimate for the 2 point boundary value problems,

Theorem 8.2; a posteriori error estimate for the 2 point boundary value problems,

Lemma 9.1; stability estimates for the dual of a general initial value problem,

Theorem 21.1; the Lax-Milgram Theorem,

Theorem 14.1; piecewise linear polynomial interpolation in 2D (give the proof only in the 1D case as we did in class for chapter 5).

Theorem 15.4; a posteriori error estimates for the Poisson's equation,

Lemma 16.5; a discrete strong stability estimate for the heat equation.


Extra assignment for GU students:

Students from GU have to do problems 2 and 24 from Övningsexemple i PDE1 TM. These should be handed in not later than at the exercise on Thursday week 6 (Nov. 30). This extra work is because you get 5 credit points.



Weekly program:

(Chapters in parantheses will be covered ''partially'' )

$\circ$ Week 1: Chapters (5), 6 and (7)

$\circ$ Week 2: Chapter 8

$\circ$ Week 3: Chapter 9

$\circ$ Week 4: Chapters 21 and 10

$\circ$ Week 5: Chapters 14 and 15

$\circ$ Week 6: Chapters 16 and 17

$\circ$ Week 7: Chapters 18 and (19)



Suggestions for exercises

(Some of these exercises will be demonstrated)

The even problems from Övningsexemple i PDE1 TM,

AND

Chapter 5: 5.11, 5.12, 5.17, 5.23, 5.27, 5.56

Chapter 6: 1: Give a varitional formulation of -u''+u=f in (0,1), with u(0)=u(1)=0.

2: Write a FEM-formulation with piecewise linear, continuous functions, and a uniform stepsize h=1/4.

3: The same as above, but with piecewise quadratic functions.

Chapter 7: 7.3, 7.5, 7.24, 7.31 (prove in addition that there is exactly one minimum), 7.54

Chapter 8: 8.1, 8.3, 8.6, 8.7, 8.8, 8.11, 8.12, 8.15, 8.16, 8.18, 8.21, 8.22, 8.23, 8.32, 8.38, 8.41

Chapter 9: 9.4, 9.5, 9.7, 9.9, 9.10, 9.12, 9.13, 9.14, 9.19, 9.22, 9.23, 9.24, 9.26, 9.27, 9.28, 9.33, 9.43, 9.45, 9.46

Chapter 21: 21.1, 21.2, 21.3, 21.4, 21.5, 21.8, 21.9, 21.13

Chapter 14: 14.7, 14.10

Chapter 15: 15.5, 15.9, 15.11, 15.15, 15.20, 15.22, 15.24, 15.35, 15.39, 15.44, 15.45, 15.47

Chapter 16: 16.4, 16.7, 16.11, 16.14, 16.15, 16.18, 16.20

Chapter 17: 17.4, 17.8, 17.9, 17.10, 17.11, 17.13, 17.17, 17.18, 17.19, 17.20, 17.33

Chapter 18: 18.1, 18.3, 18.4, 18.5, 18.6, 18.9


What has changed so far:


$\circ$ I will emphasize 1D cases

$\circ$ An example note-book,Övningsexemple i PDE1, TM, consisting of problems from some previous exams and solutions to the Lösningar för udda uppgifter i Övningsexemple i PDE1 TM, odd problems in Övningsexemple i PDE1, are now available in the cousre homepage (ps-files) and will be distributed in second week .

$\circ$ There will be a course diary describing the current process of lectures/ exercises.


Mohammad Asadzadeh
December 06/2000