代写Math 362, 2016. Assignment 2: Conservation laws (Correc

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  • 代写Math 362: Conservation laws (Corrected)
    Math 362, 2016. Assignment 2: Conservation
    laws (Corrected)
    1. Let  = (x; t), x 2 [0;1) and t  0 and let the initial condition be
    (x; 0) = f(x) =
    (
    1 ; x 2 [2; 3]
    0 ; otherwise
    Furthermore, let the boundary condition be
    (0; t) = g(t) =
    (
    1=2 ; t 2 [2; 4]
    0 ; otherwise
    Let the density satisfy the advection equation with v  3.
    (a) State the density in the general form (in terms of f and g).
    (b) Derive the density (x; t) at time t = 1.
    (c) Derive the density (x; t) at time t = 6.
    2. Consider the linear velocity model
    v() = vm
    
    1