Thursday, 16 July 2015

PDEs time again: the Transport equation

Apparently, it looks like I’m all focused on PDEs at this time of the year. There’s a good reason though! I find PDEs extremely fascinating in that they allow you to model complex behaviours of objects such as waves propagating and strings vibrating and so on. If not for their beautiful mathematical content you must love them for their applications in physics!
Take the transport equation, imagine you’d like to model the behaviour of an ocean wave. Well, in an ideal world, once generated, our wave would go on forever at a constant speed, say k. In order to find the function that represents our wave at each point at every time t, one should solve the following problem


It turns out solutions are all in the form of:


For those of you already familiar with the wave equation, this should look extremely familiar. It is indeed the equation of a wave travelling towards the positive x at speed k
If we choose a Gaussian function for instance, then we would obtain the following:


and by running the same code we used for the heat equation, here is what we get:

Since the code is pretty much the same as the heat equation I will not post it here. However, it is downloadable via dropbox by clicking here if you’d like to check it out.
Now a question should arise. Our wave is an ideal wave, it does not take into account friction and if you think of an ocean wave, it dissolves itself quickly as it meets the beach. Can we model this with the transport equation? Sure! We can use the transport equation with exponential decay. The problem we need to solve now looks a little different:


where gamma is a constant which relates to the decay rate. The general solution now looks something like this


Note that it decays and tends to zero as time tends to infinity and that it is still a wave travelling ‘to the right’. If we then use the same Gaussian as before, we obtain a much more realistic wave. Here is a short clip generated using the code at the bottom of the page (the animation is slower since I set dt=0.01 for a better resolution).

You can play with the parameters I set and see how the behaviour of the wave changes!

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