THE (UNFINISHED) PDE COFFEE TABLE BOOK

2. Laplace equation in 2D...harmonic functions from analytic functions
3. Heat equation...diffusion and smoothing
3. Backward heat equation...ill-posed problems and regularisation
3. Constant coefficient linear equations...Fourier analysis and boundedness
7. Laplace equation...the basis of potential theory
7. Biharmonic equation...solid mechanics and creeping flow
8. Minimal surface equation...mathematisation of a soap film
9. Capillary surface equation...where gases, liquids and solids meet
15. Wave equation...propagation of light and sound
15. Wave equation in 1D...left-going and right-going characteristics
17. Helmholtz equation...waveguides and scattering
19. One-way wave equations...pseudodifferential operators & absorbing boundaries
21. Beam equation...waves in an elastic rod
21. Time-dependent Schrodinger equation...probability waves and tunneling
23. Gray-Scott equations...stripes and spots, waves and rolls
24. KdV equation...solitons and inverse scattering
26. Sine-Gordon equation...wave propagation with nonlinear forcing
29. Compacton equations...approximate solitons with compact support
29. Blow-up equation with u^p nonlinearity...blow-up in finite time
29. Boussinesq equation...can an ill-posed equation be physical?
31. Blow-up equation with e^u nonlinearity...thermal runaway
31. Lewy equation...a smooth PDE with no solutions
34. Advection-diffusion equations...asymmetry and boundary layers
35. Porous medium equation...nonlinear diffusion with sharp interfaces
37. Fisher-KPP equation...nonlinear traveling waves
38. Allen-Cahn or bistable equation...exponentially long transients and metastability
42. Eigenfunctions of the Laplace equation...can one hear the shape of a drum?
45. Cahn-Hilliard equation...phase separation and coarsening
46. Perona-Malik equation...image sharpening by anisotropic diffusion
47. Kuramoto-Sivashinsky equation...a PDE with chaotic solutions
51. Inviscid Burgers equation...shocks and rarefactions
51. Complex Ginzburg-Landau equation...a laboratory for turbulence
58. Equations of fluid mechanics...half the world of PDEs
59. Klein-Gordon equation...an equation both hyperbolic and dispersive