I find it really interesting how useful this property seems to be. As I understood it:
In a continuous setting the mean value property for Laplace says that each value is the average of all its neighbors at the same fixed distance away. In our discrete setting, we can use this property and look up the four neighbors one unit away using our grid. And then since the update step for solving the heat and wave equations uses the Laplacian, we leveraged this property to solve the three classes of PDEs in this lecture (!)
keenan
@bpopeck Yep, you got it! If you understand the mean value property, you understand a lot about the Laplacian---and in turn, a lot about PDEs.
I have a whole lecture about the Laplacian here if you want to know more.
I find it really interesting how useful this property seems to be. As I understood it:
In a continuous setting the mean value property for Laplace says that each value is the average of all its neighbors at the same fixed distance away. In our discrete setting, we can use this property and look up the four neighbors one unit away using our grid. And then since the update step for solving the heat and wave equations uses the Laplacian, we leveraged this property to solve the three classes of PDEs in this lecture (!)
@bpopeck Yep, you got it! If you understand the mean value property, you understand a lot about the Laplacian---and in turn, a lot about PDEs.
I have a whole lecture about the Laplacian here if you want to know more.