Consider the one-way wave equation ut + uz = 0 on the interval (-1,3) and for t> 0 with the following two sets of initial conditions: 1- if |2 <1, u(2,0) = (5) otherwise, and u(1,0) = e-52. (6) (a) Use the forward-time forward-space me: ,n+1 +-u + 0, k h with right-point boundary condition unti where IM = 3 to compute an approx- imation to the solution at several (up to 40) time steps. Use h = 0.02 and 1 = k/h = 0.8. Demonstrate numerically (plot the solution) the instability of the scheme and show that the instability appears sooner with the less smooth initial data.
(b) Comment on the localization of the onset of instability for initial data (5) and give an estimate of the expected growth rate of the instability per time step.
(c) Using the left boundary condition u(-1) = 0, write a stable scheme and compute the corresponding approximation for data sets (5) and (6). Plot the approximations at representative time steps. Use again h = 0.02 and 1 = k/h=0.8. =

Two firms, a and b, each currently emit 100 tons of chemicals into the air. the government has decided to reduce the pollution and from now on will require a pollution permit for each ton of pollution emitted into the air. the government gives each firm 40 pollution permits, which it can either use or sell to the other firm. it costs firm a $200 for each ton of pollution that it eliminates before it is emitted into the air, and it costs firm b $100 for each ton of pollution that it eliminates before it is emitted into the air. after the two firms buy or sell pollution permits from each other, we would expect that firm a will emit a. 100 fewer tons of pollution into the air, and firm b will emit 20 fewer tons of pollution into the air. b. 20 more tons of pollution into the air, and firm b will emit 100 fewer tons of pollution into the air. c. 50 fewer tons of pollution into the air, and firm b will emit 50 fewer tons of pollution into the air. d. 20 fewer tons of pollution into the air, and firm b will emit 100 fewer tons of pollution into the air.