The Lighthill-Whitman-Richards Model. The following model for traffic flow on a one-lane road was proposed by Lighthill and Whitman in 1955 and by Richards in 1956. + g'(u). == at u(x, 0) == 0; f(x), where g(u) = u(1-u), and is the initial traffic density. (a) Assume the initial traffic density in (2) is given by f(x)= 1 2' if x < 0; 0, if x 0. (2) (3) Describe the traffic situation modeled by this initial condition in the initial- value problem (IVP) in (2). (b) Sketch the characteristic curves for the IVP in (2) with initial condition given in (3). (c) Explain why the method of characteristic curves does not yield a solution, u(x,t), of the IVP in (2) with initial condition in (3) for 0 < x 0. (d) For each > 0, define f: R R by if x < 0; f(x)= 2' E-X 2' if 0 < x < ; 0, (4) if xe. Use the method of characteristic curves to compute a solution of the IVP It +g' (u) x 0: u(x, 0) == f(x), (5) where f is given in (4) and g(u) = u(1 u). (e) Use the result of part (d) to obtain a solution of the IVP in (2) where f is given in (3). Explain the reasoning leading to your answer.