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ME565 Lecture 8: Heat Equation: derivation and equilibrium solution in 1D (i.e., Laplace's equation)
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Classroom Contents
Engineering Mathematics
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- 1 ME564 Lecture 1: Overview of engineering mathematics
- 2 ME564 Lecture 2: Review of calculus and first order linear ODEs
- 3 ME564 Lecture 3: Taylor series and solutions to first and second order linear ODEs
- 4 ME564 Lecture 4: Second order harmonic oscillator, characteristic equation, ode45 in Matlab
- 5 ME564 Lecture 5: Higher-order ODEs, characteristic equation, matrix systems of first order ODEs
- 6 ME564 Lecture 6: Matrix systems of first order equations using eigenvectors and eigenvalues
- 7 ME564 Lecture 7: Eigenvalues, eigenvectors, and dynamical systems
- 8 ME564 Lecture 8: 2x2 systems of ODEs (with eigenvalues and eigenvectors), phase portraits
- 9 ME564 Lecture 9: Linearization of nonlinear ODEs, 2x2 systems, phase portraits
- 10 ME564 Lecture 10: Examples of nonlinear systems: particle in a potential well
- 11 ME564 Lecture 11: Degenerate systems of equations and non-normal energy growth
- 12 ME564 Lecture 12: ODEs with external forcing (inhomogeneous ODEs)
- 13 ME564 Lecture 13: ODEs with external forcing (inhomogeneous ODEs) and the convolution integral
- 14 ME564 Lecture 14: Numerical differentiation using finite difference
- 15 ME564 Lecture 15: Numerical differentiation and numerical integration
- 16 ME564 Lecture 16: Numerical integration and numerical solutions to ODEs
- 17 ME564 Lecture 17: Numerical solutions to ODEs (Forward and Backward Euler)
- 18 ME564 Lecture 18: Runge-Kutta integration of ODEs and the Lorenz equation
- 19 ME564 Lecture 19: Vectorized integration and the Lorenz equation
- 20 ME564 Lecture 20: Chaos in ODEs (Lorenz and the double pendulum)
- 21 ME564 Lecture 21: Linear algebra in 2D and 3D: inner product, norm of a vector, and cross product
- 22 ME564 Lecture 22: Div, Grad, and Curl
- 23 ME564 Lecture 23: Gauss's Divergence Theorem
- 24 ME564 Lecture 24: Directional derivative, continuity equation, and examples of vector fields
- 25 ME564 Lecture 25: Stokes' theorem and conservative vector fields
- 26 ME564 Lecture 26: Potential flow and Laplace's equation
- 27 ME564 Lecture 27: Potential flow, stream functions, and examples
- 28 ME564 Lecture 28: ODE for particle trajectories in a time-varying vector field
- 29 ME565 Lecture 1: Complex numbers and functions
- 30 ME565 Lecture 2: Roots of unity, branch cuts, analytic functions, and the Cauchy-Riemann conditions
- 31 ME565 Lecture 3: Integration in the complex plane (Cauchy-Goursat Integral Theorem)
- 32 ME565 Lecture 4: Cauchy Integral Formula
- 33 ME565 Lecture 5: ML Bounds and examples of complex integration
- 34 ME565 Lecture 6: Inverse Laplace Transform and the Bromwich Integral
- 35 ME565 Lecture 7: Canonical Linear PDEs: Wave equation, Heat equation, and Laplace's equation
- 36 ME565 Lecture 8: Heat Equation: derivation and equilibrium solution in 1D (i.e., Laplace's equation)
- 37 ME565 Lecture 9: Heat Equation in 2D and 3D. 2D Laplace Equation (on rectangle)
- 38 ME565 Lecture 10: Analytic Solution to Laplace's Equation in 2D (on rectangle)
- 39 ME565 Lecture 11: Numerical Solution to Laplace's Equation in Matlab. Intro to Fourier Series
- 40 ME565 Lecture 12: Fourier Series
- 41 ME565 Lecture 13: Infinite Dimensional Function Spaces and Fourier Series
- 42 ME565 Lecture 14: Fourier Transforms
- 43 ME565 Lecture 15: Properties of Fourier Transforms and Examples
- 44 ME565 Lecture 16 Bonus: DFT in Matlab
- 45 ME565 Lecture 17: Fast Fourier Transforms (FFT) and Audio
- 46 ME565 Lecture 16: Discrete Fourier Transforms (DFT)
- 47 ME565 Lecture 18: FFT and Image Compression
- 48 ME565 Lecture 19: Fourier Transform to Solve PDEs: 1D Heat Equation on Infinite Domain
- 49 ME565 Lecture 20: Numerical Solutions to PDEs Using FFT
- 50 ME565 Lecture 21: The Laplace Transform
- 51 ME565 Lecture 22: Laplace Transform and ODEs
- 52 ME565 Lecture 23: Laplace Transform and ODEs with Forcing and Transfer Functions
- 53 ME565 Lecture 24: Convolution integrals, impulse and step responses
- 54 ME565 Lecture 25: Laplace transform solutions to PDEs
- 55 ME565 Lecture 26: Solving PDEs in Matlab using FFT
- 56 ME 565 Lecture 27: SVD Part 1
- 57 ME565 Lecture 28: SVD Part 2
- 58 ME565 Lecture 29: SVD Part 3
- 59 The Laplace Transform: A Generalized Fourier Transform
