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