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PDEs: General classification

PDEs: General classification. “Elliptic” Typical: LaPlace’s Eq. steady-state gravity, electrostatics. “Parabolic” Typical: Heat Eq. conduction. “Hyperbolic” Typical: Wave Eq. vibration, propagation. PDEs: discretization and computational molecules. Forward difference

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PDEs: General classification

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  1. PDEs: General classification “Elliptic” Typical: LaPlace’s Eq. steady-state gravity, electrostatics “Parabolic” Typical: Heat Eq. conduction “Hyperbolic”Typical: Wave Eq. vibration, propagation

  2. PDEs: discretization and computational molecules • Forward difference • Backward difference • Symmetric difference • Second difference • Mixed partials • Discrete Laplacian Example: Marching algorithm for heat equation

  3. Sketch of derivations: PDEs: Where do they come from? • Heat (diffusion) equation • Wave equation • solution: • traveling waves • boundary conditions • eigenmodes • characteristics • Dan Russell's excellent site

  4. PDEs(wrapping up; segue to Fourier analysis, DSP, image processing, etc.) • More on numerical solution • von Neumann stability analysis • Consistency • Example of inconsistency • Nonlinear vibration • Fermi-Pasta-Ulam-Tsingou and • the idea of a numerical experiment • Fourier modes, nonlinear PDEs, solitons

  5. Upcoming themes • Surprise in science • Fourier (modal) analysis • The Fourier transform becomes an algorithm • The FFT, O(n log n) vs. n2, changes life • Sound and images become digital

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