1. Consider the following initial-boundary value problem for the advection equation: (Out Ox(ezu) = 0 Vr E [1, 4], t ER, incoming flow rate at r = 1 at each time t is equal to sin(t), (PAO = (0'T)n Note that the flow rate of material through a point x in a velocity field v(r) at time t is equal to v(r)u(x, t) where u(x, t) is the concentration of the material at (x, t). I set this problem up in such a way that it models flow through a channel with an oscillating input. a) Find the characteristic curves of this problem. b) For arbitrary (r, t), identify the condition that determines whether the characteristic curve through (x, t) intersects the set t = 0, or the set x = 1. Hint: first, take the char curve with (To, to) = (1, 0) and on that curve, find a formula for a in terms of t, then consider what happens when the equality is replaced with an inequality in either direction. c) Find the solution u(x, t). Note that it will be written piecewise in two pieces. Note: your formula for u(x, t) should be a (piecewise) function of (x, t). d) Can you comment on the behaviour of the function t u(4, t)? e) Bonus: is the outflow rate at r = 4 the same as the inflow rate at r = 1 for large t? If it's not the same at each moment in time, is it the same on average