partial differential equations1 solve xut uux 0

Question

Partial Differential Equations

1) Solve xu_t + uu_x = 0 with u(x,0) = x. (Hint: Change variables x→x²)

2) Solve u_t + uu_x = 0 with the initial condition u(x,0)=1 for x≤0, 1-x for 0≤x≤1, and 0 for x≥1. Solve it for all t≥0, allowing for a shock wave. Find exactly where the shock is and show that it satisfies the entropy condition. Sketch the characteristics.

3) Show that the mass, momentum, and energy are constants of motion (invariants) for the KdV equation by direct differentiation with respect to time.