Partial Differential Equations: Graduate Level Problems and ...

Solve the Cauchy problem u t +uu x =0, u(x,0)= h(x). (5.1) The characteristic equations are dx dt = z, dy dt =1, dz dt =0, and Γ may be parametrized by (s,0,h(s)). x = h(s)t+s, y = t, z = h(s). u(x,y)=h(x−uy) (5.2) The characteristic projection in the xt-plane1 passing through the point (s,0) is the line x = h(s)t+s along which u has the ... ................
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