Introduction to Partial Differential Equations
Exercise Sheet 4
Ross Ogilvie 23rd September,
2024
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In this question we explore some other solutions to the initial value problem from Example 1.10. As we saw, for small the method of characteristics gives a unique solution
After , the characteristics begin to cross and so the method cannot assign which value should have at a point . However, we could still arbitrarily decide to choose a value of one characteristic. Consider therefore
In this question we look at an equation similar to Burgers’ equation that describes traffic. Let measure the number of cars in a given distance of road, the car density. We have seen that should be interpreted as the flux function, the number of things passing a particular point. When there are no other cars around, cars travel at the speed limit . When they are bumper-to-bumper they can’t move, call this density .