Using the methods of characteristics for a linear partial differential equation, e.g. $$u_t + au_x = f,$$
can there be a noncontinuous solution, i.e. is there any example where the characteristics do intersect? If not, why not?
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You mean a PDE of the form $u_t+au_x = f$ ? – Thomas Feb 19 '13 at 18:48
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Yes! I corrected this in my posting. – robertmiles Feb 19 '13 at 18:58
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Given the PDE $$ u_t +au_x = f$$ Note that, as $u=u(x,t) =u(x(t),t))$ it follows $$\frac{d}{dt} u = u_x \frac{dx}{dt} + u_t$$ comparing this with the PDE we get \begin{align} &\frac{dx}{dt} = a \\ & \frac{du}{dt} = f \end{align} This means the characteristic is described via $\frac{dx}{dt}=a$ which is equivalent $x = at+x_0$. Along this characteristic,$u$ changes with $\frac{du}{dt}=f$.
But as the characteristic is described via $x=at+x_0$ it is impossible to get intersecting charachteristics, as all have the same slope $a$.
Thomas
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