One dimensional boundary value problems showing interesting behaviour

Question

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?

Answer 1

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

Answer 2

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).