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Uniqueness of a solution

  1. Dec 9, 2009 #1
    If I have a PDE like Ux-Uy=0 and U(x,0)=f(x) when x in [0,1]. Then is there an uniqueness solution exist at point (5,1)?
    How can I explain it using characteristics lines?

  2. jcsd
  3. Dec 10, 2009 #2


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    Science Advisor

    If you only know f(x) between 0 and 1, you are going to have a problem extending the solution to x= 5!

    The "characteristic lines" are of the form x+ y= C and any solution to this equation is of the form F(x+y) where F is an arbitrary function of one variable. Since you require that U(x,0)= F(x+0)= F(x)= f(x) for x between 0 and 1, you will need to take F(x) to be f(x) between 0 and 1 but that does not define it for x+ y= 5+ 1= 6. Consider any number of functions "f(x)" which are identical between 0 and 1 but differ outside that interval.
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