Classification of 2nd order PDEs - PowerPoint PPT Presentation

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Classification of 2nd order PDEs
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Classification of 2nd order PDEs

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  1. Classification of 2nd order PDEs

  2. Classification of 2nd order PDEs

  3. Classification of 2nd order PDEs

  4. An important case: the heat equation in 1D Homogeneous state and its stability

  5. the heat equation in 1D (Fourier approach)

  6. The heat equation in 1D

  7. Green’s function solution of the heat equation

  8. The heat equation does not have travelling wave solutions

  9. Reaction-diffusion equations in 1D, for a scalar field

  10. The simplest case: the KISS model for plankton blooms

  11. Need for a saturation mechanism: the Fisher equation

  12. The equation is now nonlinear… Fourier decomposition does not help now

  13. Fisher equation: homogeneous solution (the logistic equation)

  14. the logistic equation

  15. back to the Fisher equation: travelling waves

  16. dynamical systems in a 2d phase space

  17. dissipative and conservative dynamical systems

  18. types of bifurcation: normal form

  19. types of bifurcation: normal form

  20. types of bifurcation: normal form

  21. Fisher equation: travelling wave

  22. Fisher equation: phase-plane picture

  23. Fisher equation: travelling wave c ≥ 2 z http://commons.wikimedia.org/wiki/File:Travelling_wave_for_Fisher_equation.svg

  24. Fisher equation: travelling wave general initial condition

  25. Fisher equation: travelling wave asymptotic form stability of the travelling wave

  26. A different form of nonlinearity… Again, Fourier decomposition does not help

  27. Heuristic, just to understand… This is an hyperbolic equation Singularity in finite time

  28. The Burgers equation Shock solutions

  29. the Cole-Hopf transformation

  30. The Burgers equation: shock structure general solution and confluence of shocks test for numerical methods stochastically-forced Burgers eq.

  31. The Korteweg-de Vries equation Travelling wave solutions: “solitary waves” on the periodic: “cnoidal waves”

  32. The inverse scattering transform for the Korteweg-de Vries equation