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Unformatted text preview: Assignment 2 ( SCI 3510 Mathematical and Statistical Modelling ) The Greenshield’s model is adopted for the velocitydensity relation without special notification in the following problems. (1) Suppose that the initial density is given by ρ ( x, 0) = 3 4 ρ opt x ∈ (∞ , 1] 2 x 4 ρ opt x ∈ [ 1 , 1 ] 1 4 ρ opt x ∈ [ 1 , + ∞ ) (a) Sketch the characteristic lines for the above initial density. (b) Give the density ρ ( x, t ) and its graph at time t = 1 and 2. (2) Suppose that the initial density is given by ρ ( x, 0) = 1 4 ρ opt x ∈ (∞ , 1] 2 + x 4 ρ opt x ∈ [ 1 , 1 ] 3 4 ρ opt x ∈ [ 1 , + ∞ ) (a) When and where a shock firstly occurs? (b) Sketch the characteristic lines for the above initial density. (c) Give the density ρ ( x, t ) and its graph at time t = 1 4 t s , 1 2 t s , t s , and τ &gt; t s ....
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 Spring '11
 wong

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