Partial Differential Equations

1) Solve xu_{t} + uu_{x} = 0 with u(x,0) = x. (Hint: Change variables x→x^{2})

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.

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