ME565 Lecture 12
Engineering Mathematics at the University of Washington

Fourier Series

Notes: faculty.washington.edu/sbrunton/me565/pdf/L12.pdf
Matlab code: faculty.washington.edu/sbrunton/me565/matlab/L12_Fourier.m


faculty.washington.edu/sbrunton/

1. Complex numbers and functions
2. Roots of unity, branch cuts, analytic functions, and the Cauchy-Riemann conditions
3. Integration in the complex plane (Cauchy-Goursat Integral Theorem)
4. Cauchy Integral Formula
5. ML Bounds and examples of complex integration
6. Inverse Laplace Transform and the Bromwich Integral
7. Canonical Linear PDEs: Wave equation, Heat equation, and Laplace's equation
8. Heat Equation: derivation and equilibrium solution in 1D (i.e., Laplace's equation)
9. Heat Equation in 2D and 3D. 2D Laplace Equation (on rectangle)
10. Analytic Solution to Laplace's Equation in 2D (on rectangle)
11. Numerical Solution to Laplace's Equation in Matlab. Intro to Fourier Series
12. Fourier Series
13. Infinite Dimensional Function Spaces and Fourier Series
14. Fourier Transforms
15. Properties of Fourier Transforms and Examples
16. Discrete Fourier Transforms (DFT)
17. Bonus: DFT in Matlab
18. Fast Fourier Transforms (FFT) and Audio
19. FFT and Image Compression
20. Fourier Transform to Solve PDEs: 1D Heat Equation on Infinite Domain
21. Numerical Solutions to PDEs Using FFT
22. The Laplace Transform
23. Laplace Transform and ODEs
24. Laplace Transform and ODEs with Forcing and Transfer Functions
25. Convolution integrals, impulse and step responses
26. Laplace transform solutions to PDEs
27. Solving PDEs in Matlab using FFT
28. SVD Part 1
29. SVD Part 2
30. SVD Part 3

This is the second part of Prof. Steve Brunton's course on Mechanical Engineering Mathematics. It will provide an in-depth overview of powerful mathematical techniques for the analysis of engineering systems. In addition to developing core analytical capabilities, students will gain proficiency with various computational approaches used to solve these problems. Applications will be emphasized, including fluid mechanics, elasticity and vibrations, weather and climate systems, epidemiology, space mission design, and applications in control.

In this course, we will develop many powerful analytic tools. Equally important is the ability to implement these tools on a computer. The instructor and TAs use Matlab, and all examples in class will be in Matlab.