Math 4Q03  Numerical Methods for Ordinary and Partial Differential Equations  Winter 2000 McMaster University DR. N. KEVLAHAN Office: BSB-246E, Tel: x23412 Email: kevlahan@mcmaster.ca Web: icarus.math.mcmaster.ca/kevla Office hours: Monday 14:30-15:30, Thursday 13:30-14:30 and by appointment

Purpose of the course

Most mathematical problems that arise from applications in science and engineering are difficult or, more often, impossible to solve using techniques from algebra or analysis courses. However, approximate solutions are almost always possible using numerical techniques implemented on a computer. The convergence and stability of these methods can be calculated and thus the solution can be found to a specified precision.

The objective of this course is to teach a variety of numerical methods (e.g., spline interpolation, numerical integration and differentiation, numerical linear algebra and root finding) by applying them to the solution of ordinary and partial differential equations. This approach shows how these methods can be combined to solve a particular practical problem: the solution of ODE's and PDE's.

Outline

An introduction to numerical methods and their application to the numerical solution of initial and boundary value problems of ODEs and PDEs. More specifically, topics to be covered include:

1.

Introduction to numerical methods: interpolation, numerical differentiation and integration.

2.

Ordinary differential equations.

• Initial value problems: Euler's method, predictor-corrector, Runge-Kutta methods, linear multistep, convergence and stability, techniques for stiff equations.
• Boundary value problems: finite differences, Galerkin method, collocation, finite elements (interpolation and splines).
• Numerical solution of algebraic equations: iterative methods, Jacobi, Gauss-Seidel, SOR, Newton's method.

3.

Partial differential equations:

• Introduction to PDE's: classification and characteristics.
• Parabolic equations: spatial discretization by finite differences, explicit and implicit methods for time stepping, numerical instabilities and physical interpretation.
• Convergence theory: consistency, stability and convergence, Lax's equivalence theorem.
• Hyperbolic equations: finite differences, convergence analysis.
• Elliptic equations (steady-state equations): finite differences and finite element formulations.

Texts

The primary texts for the course are:

1.

Mathews, J. H. & Fink, K. D. 1999 Numerical methods using MATLAB (3rd edition). Prentice Hall.

2.

The Student edition of MATLAB, Version 4 for Microsoft Windows, Prentice Hall.

Optional texts include:

1.

Pratap, R. 1999 Getting started with MATLAB 5. Oxford University Press.

2.

Press, W. H., Flannery, B. P., Teukolsky S. A. & Vetterling, W. T. 1992 Numerical Recipes in C/Fortran, Cambridge University Press, (QA76.783.C15N865, QA 297.N866).

3.

Sewell, G. 1988 The Numerical solution of Ordinary and Partial Differential equations. Academic Press (QA 372.S4148).

Assignments

Five problem sheets will be given and marked for credit. Assignments are to be dropped into the Math 4Q03 locker in the basement of the Burke Sciences Building by 15:00 on the due date. No late assignments will be accepted. Solutions to assignments and tests will be put on reserve in the Thode Library. In addition, there will be one project where a single problem is investigated in depth.

Assignment given Assignment due 

January 7                January 21 

January 21               February 4 

February 4               February 21

March 10                 March 27 

March 24                 April 7

Project given Project due 

February 18              March 20

Tests

There will be one 50 minute test, held in a location to be announced, during the regularly scheduled class hour on the following date: Monday 28 February.

Summary of topics for test

Summary of topics for test (pdf)

Review session for test: Friday 18 February 13:30-15:20 at my office
Test room: A-L  PC B127 (normal room)
                      M-Z BSB B154

Special Office hour: 9:30-10:30 Friday 25 February.

Final exam

There will be a three-hour final examination during the April examination period.

Summary of topics for Final Exam (pdf)

Final mark

I have posted course marks on my door (BSB 246E).  Please check your marks and report any problems to me before the final exam.

The final mark will be calculated as follows:

Assignments	20%
Project	10%
Test	20%
Final exam	50%

I reserve the right to change the weight of any portion of this marking scheme. If changes are made, your grade will be calculated using the original weightings and the new weightings, and you will be given the higher of the two grades. At the end of the course the grades may be adjusted but this can only increase your grade and will be done uniformly. I will use the grade equivalence chart in the university calendar to convert between letter grades, grade points and percentages.

MATLAB Information

• MATLAB is installed on the computers in the lab on the 2nd floor of the Burke Science Building. A package is also available on diskettes bound in the book The student edition of MATLAB.
• More information about MATLAB can be found at The Math Works Inc. Web site www.mathworks.com. The book Getting started with Matlab 5 is a good guide to programming in MATLAB.

Statement on academic ethics

Attention is drawn to the ``Statement on Academic Ethics'' and ``Senate Resolutions on Academic Dishonesty'' as found in the Senate Policy Statements distributed at Registration and available in the Senate Office. Any student who infringes on one of these resolutions will be treated according to the published policy. In particular, it is expected that the assignments and project shall be done and submitted as individual work. Students may discuss general problems or approaches, but the final solution must be a result of the student's own effort.

The Faculty of Engineering is concerned with ensuring an environment that is free of all adverse discrimination. If there is a problem that cannot be resolved by discussion among the persons concerned, individuals are reminded that they should contact their Department Chair, the Sexual Harassment Office or the Human Rights Consultant, as soon as possible.

Assignments

 Assignment 1
 Assignment 2
 Assignment 3
 Assignment 3 (pdf)
 Assignment 4 (pdf)
 Assignment 5 (pdf)

 Project
 Project (pdf)

 McMaster University Mathematics and Statistics Department Home Page

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