THE (UNFINISHED) PDE COFFEE TABLE BOOK



Lloyd N. Trefethen and Kristine Embree, editors

Unpublished, 2001

During 2000-2001 a group project based in the Oxford University was begun to write this book. The vision was 100 2-page spreads, each one giving exactly the most useful possible starting information about a different partial differential equation, with beautiful color illustrations. Many people at Oxford and around the world contributed drafts, which were then extensively rewritten and edited to help bring about a uniform style and mathematical level. In the end, 34 of the pieces were brought into reasonably finished form.



Many people have asked since then about the status of the PDE Coffee Table Book. Unfortunately, we have had to tell them that the project is stalled, with no plans at present to complete it.



As people keep asking, it seems unreasonable not to make this easily available. Thus, with pride for what we finished and regret for what we didn't, here are the 34 finished pages. The numbers (which are not all distinct) give some idea of where each page might have ended up in the finished book. These may be used freely for educational purposes, but all publication rights are reserved. Copyright Lloyd N. Trefethen, 2011.





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





Nick Trefethen

Oxford University Mathematical Institute

24-28 St. Giles

Oxford OX1 3LB, UK

trefethen@maths.ox.ac.uk