### Lecture Description

ME565 Lecture 8

Engineering Mathematics at the University of Washington

Heat Equation: derivation and equilibrium solution in 1D (i.e., Laplace's equation)

Notes: faculty.washington.edu/sbrunton/me565/pdf/L08.pdf

faculty.washington.edu/sbrunton/

### Course Index

- Complex numbers and functions
- Roots of unity, branch cuts, analytic functions, and the Cauchy-Riemann conditions
- Integration in the complex plane (Cauchy-Goursat Integral Theorem)
- Cauchy Integral Formula
- ML Bounds and examples of complex integration
- Inverse Laplace Transform and the Bromwich Integral
- Canonical Linear PDEs: Wave equation, Heat equation, and Laplace's equation
- Heat Equation: derivation and equilibrium solution in 1D (i.e., Laplace's equation)
- Heat Equation in 2D and 3D. 2D Laplace Equation (on rectangle)
- Analytic Solution to Laplace's Equation in 2D (on rectangle)
- Numerical Solution to Laplace's Equation in Matlab. Intro to Fourier Series
- Fourier Series
- Infinite Dimensional Function Spaces and Fourier Series
- Fourier Transforms
- Properties of Fourier Transforms and Examples
- Discrete Fourier Transforms (DFT)
- Bonus: DFT in Matlab
- Fast Fourier Transforms (FFT) and Audio
- FFT and Image Compression
- Fourier Transform to Solve PDEs: 1D Heat Equation on Infinite Domain
- Numerical Solutions to PDEs Using FFT
- The Laplace Transform
- Laplace Transform and ODEs
- Laplace Transform and ODEs with Forcing and Transfer Functions
- Convolution integrals, impulse and step responses
- Laplace transform solutions to PDEs
- Solving PDEs in Matlab using FFT
- SVD Part 1
- SVD Part 2
- SVD Part 3

### Course Description

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.