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I want to show some users of a piece of software some solutions of one-dimension boundary value problems (can also be initial value problems). I'm after a collection of problems whose solutions are very interesting or the BVP are of a particularly interesting nature. The problems can be linear or non-linear. Does anyone have any suggests?

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Wiki-ized: this is a question looking for examples. –  Willie Wong Jan 28 '11 at 18:31
    
What do you mean by 1D BVP? Is it only ODEs or PDEs in 1D space + time that you are talking about? In the latter case, it is clearer to call them initial-boundary value problems. –  timur Jan 29 '11 at 12:48
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This(site about boundary value problems) should be useful. Some of the interesting boundary value problems they list are the following:

  • Heat Equation
  • Wave Equation
  • Laplace Equation
  • Poisson Equation
  • Schrodigner's Equation
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In terms of the initial value problem, a large class of interesting 1-dimensional questions arise from studying various approximations to surface waves starting from fundamental equations of fluid dynamics.

The Korteweg-de Vries equation and its cousins gKdV and mKdV all lead to demonstrations of a competition between cohesive self-interaction that leads to soliton formation, and dispersive wave-like phenomenon.

(What is especially interesting is to look at the purely dispersive part of the KdV equation $$ \partial_t \phi + \partial_x^3 \phi = 0 $$ and the purely transport part of the equation $$ \partial_t \phi + \phi\partial_x\phi = 0 $$ and note their characteristic behaviours. Then compare it against various different initial data for the KdV equation.)

You can also consider looking at the Benjamin-Bona-Mahony equation, which shows off some long wavelength behaviour (in constrast to the KdV solitons, which tends to be more spatially concentrated).

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Thank you. Great! –  alext87 Jan 29 '11 at 18:04
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