Consider the hyperbolic system ut uv uradicv and u v gt 0

Question

Assignment Questions-

Q1. Consider the equation

u_t + (u⁴/4)x = 0

Find the admission solution for each of the following initial data:

Q2. Consider the hyperbolic system

u_t + uv= = u√v,

v_t + vu_x = v√u,

and u, v > 0. Determine the characteristics and the corresponding ODEs. Show that along one of the characteristics, the following Riemann invariant holds

√u - √v = const.