Question: Consider the nonlinear transport equation u t + c(u)u x = 0; x R ; t > 0 with initial condition u(0; x) = f(x).

 Consider the nonlinear transport equation

ut + c(u)ux = 0; xR; t > 0

with initial condition u(0; x) = f(x).

1. Show that if c and f are both increasing or both decreasing, then there is no shock.

2. Using c(u) = u and

f(x)=1,x<01+x,0<x<12,x>1

find the solution. Draw the characteristics.

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