Question

Suppose a factory spills pollutants into a river; assume the pollutants flow downstream at a speed of c. Let u(t, x) model the concentration of the pollutants at time t at a distance x from the factory. u(t, x) satisfies the transport equation:

∂u/∂t + c ∂u/∂x = 0

-Let u(t, x) = u(t, z + ct) = w(t, z). Rewrite the transport equation as a differential equation in w.

-Explain why your work implies w(t, z) = F(z), and consequently u(t, x) = F(x − ct)

-Suppose u(0, x) = 100e −x . Solve the transport equation by finding the function u(t, x).

-Suppose u(t, 0) = 50t 100+t . Solve the transport equation by finding the function u(t, x).

Answer 1