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Partial DIfferential Equations problems

  1. Oct 5, 2005 #1
    Here is one of them - i posted it in another thread and i am getting help in there https://www.physicsforums.com/showthread.php?t=91781

    this is another of my problems
    Show that if C is a piecewise continuously differentiable closed curve bounding D then the problem
    [tex] \nabla^2 u= -F(x,y) \ in\ D[/tex]
    [tex] u = f \ on \ C_{1} [/tex]
    [tex] \frac{\partial u}{\partial n} + \alpha u = 0 \ on \ C_{2} [/tex]
    where C1 is a part of C and C2 the remainder and where alpha is a positive constant, has at most one solution.

    now i know that [tex] \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = -F(x,y) [/tex]

    now im not quite sure how to connect the C1 part to C2 part...
    would it be something liek C= C1 + C2?

    but how would one go about showing that this has at most ONE solution?? I m not quite sure how to start ... Please help

    another one
    Show that the problem
    [tex] \frac{\partial}{\partial x} (e^x \frac{\partial u}{\partial x} + \frac{\partial}{\partial y} (e^y \frac{\partial u}{\partial y} = 0 \ for \ x^2+y^2 < 1 [/tex]
    u = x^2 for x^2 + y^2 = 1
    has at most one solution
    Hint Use the divergence theorem to derive an energy identity


    Perhaps i dont remember a theorem i should have learnt in ap revious class... or i am not familiar with it but what would i use the divergence theorem here?
    i eman i can get it down to this
    [tex] e^x \frac{\partial}{\partial x} (u + \frac{\partial u}{\partial x}) + e^y \frac{\partial}{\partial y} (u + \frac{\partial u}{\partial y}) = 0 [/tex]
    but hereafter i am stuck, please do advise!

    Thank you!
     
    Last edited: Oct 5, 2005
  2. jcsd
  3. Oct 5, 2005 #2
    ok so i can rewrite the second euqation as

    [tex] e^x \frac{\partial}{\partial x} (u + u_{x}) + e^y \frac{\partial}{\partial y} (u + u_{y}) = 0 [/tex]

    also is [tex] u+ u_{x} [/tex] written as something else...
    how would i apply the divergence principle here?
     
  4. Oct 6, 2005 #3
    can anyone help me with this!
     
  5. Oct 6, 2005 #4

    Physics Monkey

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    Science Advisor
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    For the first problem, you might begin by assuming that two solutions exist which satisfy the differential equation and boundary conditions. The difference of the two solutions satisfies a simpler set of equations, right? Maybe this is a good place to start.

    For the second problem, the original equation already looks like the divergence of a vector field in 2d. Maybe you should start from this observation.
     
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