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fThere is something I do not understand about the Goursat problem:

  1. For a first order PDE if you prescribe the a value at a point, you can propagate the solution along a charateristic.
  2. For a second order hyperbolic PDE you need prescribe the function and normal derivative. In this case can you also propage the solution along a charateristic? If so, along which charateristic is it (there are two charateristic curves passing through each point).
  3. Finally, the Goursat problem involves prescribing the function along two intersecting charactristics. But if (2) makes any sense then if you know the function and normal derivative at a point, then you know the solution along a characteristic, and then you have the Goursat data and can solve. So knowing the function and derivative at one POINT is enough. I know this does not make sense, but where is the error??

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Let's consider the 1D wave equation $u_{tt}=c^2u_{xx}$ with zero normal derivative at the beginning. The solution is really simple: $u(x,t)=\frac{1}{2}(f(x+ct)+f(x-ct))$ where $f(x)=u(x,0)$ is the initial value. Through each point $(a,0)$ there are two characteristic lines $(x\pm ct,t)$, along with the solution propagates. But at each point of either characteristic there is another one, with the opposite slope, which adds its own contribution. As a result, you can't find the values of $u(a+ ct,t)=\frac{1}{2}(u(a)+u(a+2ct))$ without knowing the initial values elsewhere.