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

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