assignment1[2]

assignment1[2] - Assignment 1 SCI 3510 Mathematical and...

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Assignment 1 ( SCI 3510 Mathematical and Statistical Modelling ) (1) Prove the equation of state for variable velocity u ( x, t ) and density ρ ( x, t ). (2) Suppose that the velocity u ( x, t ) is selected as the prime variable. Find the relation between flux q ( x, t ) and velocity u ( x, t ) for Greenshields model. Then prove the identity: U max parenleftbigg ρ - ρ 2 ρ max parenrightbigg = ρ max parenleftbigg u - u 2 U max parenrightbigg , (3) It is officially suggested by the Transportation Department of US to keep three car-length apart from the front car for a safety drive. Show that the maximum flux reaches 3 4 q max for Greenshields model, if the principle of three car-length is obeyed. (4) Suppose the velocity-density relation is modelled as u ( ρ ) = U max parenleftbigg 1 - ρ 2 ρ 2 max parenrightbigg , Give a brief discussion with comparison to Greenshields model ( 300 words limited ). Solution From the given velocity-density relation, we have ( i ) u = U max if ρ = 0 ( ii ) u = 0 if ρ = ρ max ( iii ) ∂u ∂ρ = - 2 U max ρ ( ρ max ) 2 0 Then the tree principles in modelling the velocity-density relation are satisfied. This model is acceptable.
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