4. (Week 4) The transport of silt along a riverbed satisfies the equation Ut + (1 – 2?)ừy = 0 where u(t, x, y) is the de

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4. (Week 4) The transport of silt along a riverbed satisfies the equation Ut + (1 – 2?)ừy = 0 where u(t, x, y) is the density of silt, y E R is the direction of the river flow, and x € (-1,1) is the direction to the other bank. (a) The variable x can be treated as a constant when solving this PDE. Explain why this is the case. (b) State the characteristic equations of the above equation (c) Determine the characteristic paths for this equation. Sketch them when x = 0.5. (d) A truck drove over the riverbed at t = 0, mixing up the bed. This gave the system the initial distribution 1 -1 < x,y< 1 0 otherwise a(0, 2, 3) = Solve for the u(t, x,y). (e) Sketch the distribution of silt across the river at t = 0.5, 1, 2.