Problem 5 (15 points) Solve the initial boundary value problem on the first quadrant: Ut + Ux = 0, x > 0,t>0, u(x,0) = 0, x ≥ 0, u(0, t) = t², t ≥ 0. Hint: Use the geometric approach to the method of characteristics by sketching the characteristics on the x-t plane. The answer should be a piecewise function depending on where the backward characteristic emanating from (x, t) 'lands' first.

Fig: 1