# Classification of 2nd order PDEs - PowerPoint PPT Presentation

Classification of 2nd order PDEs

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Classification of 2nd order PDEs

## Classification of 2nd order PDEs

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##### Presentation Transcript

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