Question

5. (5 pts) Traffic can be modeled using the conservation law ut + z = 0 for the density of cars u, where o (u) is the flux function. The flux defines the average velocity of traffic cu via o = uc. A reasonable model may be alu) - cu (1 - (32)). c(u) = Cmax 1- Umax Derive the strong form of the conservation law (i.e., the PDE).

Answer 1

Solution: Given that: The con seovation law + De=0 The density of cars le The fleez feinction peu) The average velocity of traffic Clu) via 0 = cc. - - -- lolla o.za K 0.60K h = a tak bi = 0.29tt by = @ b3 = 0.6 @+t y = 2a Ya = atat cs Scanned. Y = 2130 2

Let the trapezoidal Section which y devided into three paëls 1, 2, and 3. o will find the centooid position: 0.29 ++ +@+ C0.6a+t) 0.8 a tatta 1.8 atat ga AY + A2 Y + A3 Y3 ATA2 TA3 Š x ©. 2a+t) xCatat) x 23a + Qxcatat) x Q 42€ Ź C0.6att)?gax Carpt) Š x 10.2a +t) * Catat) + acatat)+ į co.batt) Catat) 6.2a++) (0+2ba aca+9+) Catzt) O•catt) ca+24) + 3 + а (0.2a+t)(a+2t), 2(a+2t) 40.60+t)Cat2t) This using the formula given below Scanned with CamScanner

( xb, xh)y, + baxhxy + ( x bəxh) Y3 (1xbixh) + baxha 1 / 2 x 63 xh) CS Scanned with a CamScanner