Question: 1. Show that UN(x, t) = sinNxe N't solves the diffusion equation for all N > 0 and for all t and for all x.

1. Show that UN(x, t) = sinNxe N't solves the diffusion equation for all N > 0 and for all t and for all x. What happens to un(x, 0) as N - co? What happens to UN(x, -1) as N - co. The UN(x, 0) represent the profile of the solution "now" and the un(x, -1) is the profile one unit of time ago. Make a statement about stability (that is, note that two solutions might be close to each other for t = 0 but does that mean they are close to each other at t = -1?)

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