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Prerequisites
Calculus of Several Variables (18.02) and Differential Equations (18.03) or Honors Differential Equations (18.034)
Course Outline
This course has four major topics:
- Applied linear algebra (so important!)
- Applied differential equations (for engineering and science)
- Fourier methods
- Algorithms (lu, qr, eig, svd, finite differences, finite elements, FFT)
My Goals for the Course
I hope you will feel that this is the most useful math course you have ever taken. I will do everything I can to make it so. This will not be like a calculus class where a method is explained and you just repeat it on homework and a test. The goals are to see the underlying pattern in so many important applications—and fast ways to compute solutions.
Assignments and Exams
This course has ten problem sets, three one-hour exams, and no final exam. You may use your textbook and notes on the exams.
Grades
Let me try to say this clearly: my life is in teaching, to help you learn. Grades have come out properly for 20 years (almost all A-B). I will NOT spend the semester thinking about grades. I hope you don't either. The homeworks will be important and I plan 3 exams and no final. Those exams are open book and a chance for you to bring key ideas together.
Text
The textbook for this course is:
Strang, Gilbert. Computational Science and Engineering. Wellesley, MA: Wellesley-Cambridge Press, 2007. ISBN: 9780961408817. (Table of Contents)
Information about this book can be found at the Wellesley-Cambridge Press Web site, along with a link to Prof. Strang's new "Computational Science and Engineering" Web page developed as a resource for everyone learning and doing Computational Science and Engineering.
Calendar
Video lectures.
| LEC # |
TOPICS |
| 1 |
Four special matrices |
| R1 |
Recitation 1 |
| 2 |
Differential eqns and Difference eqns |
| 3 |
Solving a linear system |
| 4 |
Delta function day! |
| R2 |
Recitation 2 |
| 5 |
Eigenvalues (part 1) |
| 6 |
Eigenvalues (part 2); positive definite (part 1) |
| 7 |
Positive definite day! |
| R3 |
Recitation 3 |
| 8 |
Springs and masses; the main framework |
| 9 |
Oscillation |
| R4 |
Recitation 4 |
| 10 |
Finite differences in time; least squares (part 1) |
| 11 |
Least squares (part 2) |
| 12 |
Graphs and networks |
| R5 |
Recitation 5 |
| 13 |
Kirchhoff's Current Law |
| 14 |
Exam Review |
| R6 |
Recitation 6 |
| 15 |
Trusses and ATCA |
| 16 |
Trusses (part 2) |
| 17 |
Finite elements in 1D (part 1) |
| R7 |
Recitation 7 |
| 18 |
Finite elements in 1D (part 2) |
| 19 |
Quadratic/cubic elements |
| 20 |
Element matrices; 4th order bending equations |
| R8 |
Recitation 8 |
| 21 |
Boundary conditions, splines, gradient and divergence (part 1) |
| 22 |
Gradient and divergence (part 2) |
| 23 |
Laplace's equation (part 1) |
| R9 |
Recitation 9 |
| 24 |
Laplace's equation (part 2) |
| 25 |
Fast Poisson solver (part 1) |
| 26 |
Fast Poisson solver (part 2); finite elements in 2D (part 1) |
| R10 |
Recitation 10 |
| 27 |
Finite elements in 2D (part 2) |
| 28 |
Fourier series (part 1) |
| R11 |
Recitation 11 |
| 29 |
Fourier series (part 2) |
| 30 |
Discrete Fourier series |
| 31 |
Examples of discrete Fourier transform; fast Fourier transform; convolution (part 1) |
| R12 |
Recitation 12 |
| 32 |
Convolution (part 2); filtering |
| 33 |
Filters; Fourier integral transform (part 1) |
| 34 |
Fourier integral transform (part 2) |
| R13 |
Recitation 13 |
| 35 |
Convolution equations: deconvolution; convolution in 2D |
| 36 |
Sampling Theorem |