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

TMA371 Partial Differential Equations TM, 1999

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 (5 for GU-students).

Teachers:

Mohammad Asadzadeh (Lectures/Exercises)
telephone: 7723517, 3359518(home)
e-mail:mohammad@math.chalmers.se
URL:http://www.math.chalmers.se/~mohammad
Matematiskt Centrum, Eklandagatan, room 2331

Rikard Bergström (Exercises/Computer assignments)
telephone: 772 53 14, 221771 (home)
e-mail: ribe@math.chalmers.se
URL:http://www.math.chalmers.se/~ribe
Matematiskt Centrum, Eklandagatan 80, ASECO

Where and when:

Tuesdays: weeks 2, 3, 5, 6, 7 in VH, and week 4, in VF

Thursdays: weeks 1-6 in EB, and week 7, in VF

Lectures Tuesdays 13.15-15:00 and Thursdays 10.00-11.45,

Tuesdays: Exercises 15.15-16:00, Time for questions 16.15-17.00, Thursdays: Time for questions 8.00-8.45, Exercises 9.00-9.45.

The definite classroom and possible exercise groups will be announced during the first week.

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

Monday 25/10, Tuesday 26/10, Thursday 28/10, Monday 01/11

Time: 17.30-19.30; Place MD6

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 week4).

$\circ$ Assignment 1b (handed in no later than week4).

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

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

Unfortunately, we have not been able to construct the second computer assignment in Matlab. Instead we have to use a Unix-program developed here at math called Femlab. To overcome problems using this software, Rickard will schedule with you for an extra 2 hours to demonstrate Femlab and help with problems that come up.

The computer assignments:

The computer assignments use Matlab as the main tool. You can sit at MC if you want to (Rickard can give you a log-in name), but as long as you have access to Matlab 5.2 you can sit anywhere. We have not booked any computer rooms or tutorial time as we don't think it will be needed. If you don't agree, tell us and we will try to arrange something.

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)
Assignment 1b (bonus generating)
Assignment 2b (bonus generating) (no html available, only postscript)

Old exams with solutions:

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

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 14, 14.15-18.15 in VV, and also December 21, 8.45-13.45 in MD9 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,

Theorem 9.4; a priori error estimate for general initial value problems,

Theorem 21.1; the Lax-Milgram Theorem,

Theorem 14.1; piecewise linear polynomial interpolation in 2D,

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 Tuesday 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.

M. Asadzadeh, 19 Oct. 1999