Nonunique solutions of a partial differential equation

Some quasilinear partial differential equations don't have unique solutions...


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The partial differential equation  y du/dx + x du/dy = 0 with initial conditions u(0,y)=y^2 can be solved using the method of characteristisc, but it turns out that it does not have a unique solution. The solution u(x,y) = y^2 - x^2 is just one of infinitely many solutions. The characteristics show that for x > |y|, the initial conditions don't determine the solution. This model shows a family of alternative solution surfaces. Note also that if two solution surfaces intersect, they do so in a characteristic.

I printed this model in two parts on an Ultimaker 2, each part takes a bit over 25 hours. Then both parts are glued together along the x-axis.

Three files are provided: the complete model, which is not printable on a FDM printer, and two halves that can be printed. Scale them up by a factor of 45, otherwise the sheets of the surfaces are two thin.

The program that produced the model can be found on Github:



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Ecfee3a525c5fdc66265f478c804864f?default=blank&size=40Andreas Müller published this design ago