75048 Mathematical modeling 1998

Lecturer:

Harald Hanche-Olsen <hanche@math.ntnu.no>
Room 1154, Sentralbygg II, tel. 7359 3525.

Very latest news

The exam problems are here, in Norwegian and in English (in PostScript, without the misprint but including one piece of information which was missing from the original).

A suggested solution and a one-page summary are now available. Please let me know if you find any misprints or mistakes in the solution.

Assistant for the problems:

Henning S. Mortveit
Room 1202, Sentralbygg II, tel. 7359 1693.

The reading list (pensumliste) is available

Latest news

• A collection - no, three collections - of old exam problems are available from the department office
• Several people have asked if the grades for the modeling seminar reports will be available. The answer is yes, they will be, but not before the exam. (But rest assured that I do not see any chance of any of them receiving a really bad grade.) I will also try to make a more detailed evaluation of the reports available, but that may take a bit longer yet.
• I will be away from the office on 18-30 December and again 7-10 January. I expect to be here most of the remaining time.

Course form:

The course is based on lectures, problems, and one modeling seminar. The latter is done in small groups.

Lectures

Read about what has been lectured on

Problems:

Problems have been handed out, but are also available in PostScript form.

Problem:	1	2	3	4	5	6
Solution:	1	2, 2+	3	4	-	6

Problem set 1 is in Norwegian. The remaining problem sets, and all the solutions, are in English.
For problem set 2, the solution to problem 1 is in the document marked 2+. The note on similarity solutions of the heat equation is a useful addendum.
Problem set 5 is the modeling seminar problem. I did not plan to make a solution available for it, but I think I'll make a brief note with some of the essential points, together with feedback on the reports that were handed in.
There was a misprint in the last problem of the second set (corrected in the online copy): A square root was missing from the scaled equation.

Literature:

C.C. Lin & L.A. Segel, Mathematics applied to deterministic problems in the natural sciences SIAM 1988 (ISBN 0-89871-229-7).

Also, various handouts, including

• A note on Buckingham's pi-theorem
• A note on Lin & Segel's physiological flow problem
• A note on Conservation laws.
• A note on similarity solutions of the heat equation

Exam:

1999-01-11, 6 hours, written test. The reports from the modeling seminar are graded, and count with 20% towards the final grade.

Course contents and main ideas

The course aims at making the students familiar with simple, but proven techniques in mathematical modeling. The goal is to promote an analytical and critical attitude towards practical problem solution, as opposed to a `cook book mentality' and unreasoned use of software.

Whereas regular mathematics courses usually present difficult techniques applied to simple problems, in this course we take almost the opposite approach. Here, inventiveness and creativity are important factors. As in past years, we will use a modeling seminar, where students work in groups of four students per group.

Themes

Dimensional analysis

• helps to assess the reasonableness of a model and which variables it should contain
• reduces the number of variables and parameters to a minimum
• reduces the number of needed experiments, on computers as well as in the lab
• provides the fundamental theory behind experiments on scale models

Scaling

• systematic method for the analysis of problems
• forces you to make estimates and to understand the problem
• helps you understand what is important and what is not
• produces dimensionless equations with small (or large) parameters
• important to do before numerical computations are done

Perturbation methods

• yields approximate solutions of equations with small (or large) parameters
• a useful complement of numerical computations

Simple dynamical models

• commonly occuring
• easily understood

Conservation principles

• mathematical description of fundamental laws of nature
• useful in many areas

Hyperbolic first order equations

• direct from the conservation principles
• the simplest equation yielding discontinuous solutions
• require modest knowledge