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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This(site about boundary value problems) should be useful. Some of the interesting boundary value problems they list are the following:
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).