辅导案例-MATH4PD-Assignment 1

  • July 31, 2020

MATH4PD — Assignment 1 due on 7 August at 5pm For each initial value problem (IVP) considered below, use the method of characteristics to • find the general solution of the IVP (in explicit of implicit form), • draw the characteristic curves and the solution at relevant times, • create movies of the time evolution of the solution u(t, x), • discuss/analyse the main features of the solution, e.g. the effect of the parameters, the long-term behaviour, … Along with your report (preferably written in LATEX), hand in all your Matlab codes and video files. Also, please reference any published resource you may use to complete this assignment. Consider the following IVPs. (a)  ∂u ∂t + c(x) ∂u ∂x = atu, u(0, x) = e−(x+2) 2 , where c(x) = 1 1 + x2 and a is a parameter. (b)  ∂u ∂t + c(x) ∂u ∂x = b1u 2, u(0, x) = 1 2 cosx+ b2, where c(x) = e−x 2/10, and b1 ≥ 0 and b2 ∈ [−1, 1] are parameters. TURN PAGE OVER 1 (c)  ∂ρ ∂t + ∂ ∂x (uρ) = 0, ρ(0, x) = f(x), where u(ρ) = umax(1 − ρ/ρmax) and f(x) = 1 − |x| if |x| < 1 and 0 otherwise. This PDE is a model for traffic flow, where ρ describes the traffic density (i.e. average number of cars per unit length) and u is the traffic velocity, so that both ρ and u are non-negative quantities. Here, you can set umax = 1 and ρmax = 2. Further assume there is a red light at x = xr, where xr ≥ 1 is a parameter. 2

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