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Non-linear operator

  1. Sep 24, 2009 #1
    Hello,

    Could anyone help explain why

    [tex]\partial[/tex]/[tex]\partial[/tex]x[K(x,u)[tex]\partial[/tex]u/[tex]\partial[/tex]u]

    is not a linear operator?
     
  2. jcsd
  3. Sep 24, 2009 #2
    Are you sure you have that written down right?

    [tex]\frac{\partial u}{\partial u} = 1[/tex]
     
  4. Sep 24, 2009 #3
    Sorry, I meant:

    [tex]\partial[/tex]/[tex]\partial[/tex]x[K(x,u)[tex]\partial[/tex]u/[tex]\partial[/tex]x]

    I understand that

    [tex]\partial[/tex]/[tex]\partial[/tex]x[K(x)[tex]\partial[/tex]u/[tex]\partial[/tex]x]

    is a linear operator, but I do not get why making K a function of both x and u should non-linearize this operator.
     
  5. Sep 24, 2009 #4
    The best way to test quickly for a linear operator is just to plug in the definition... that is put in

    [tex]\lambda _1 u_1 + \lambda _2 u_2[/tex]

    to your operator above. Use the rules of differentiation and see what comes out. This would be a good exercise.

    It seems there might have to be some conditions on K(x, u) for the above to be non-linear. I'm out the door myself, so hope this helps.
     
  6. Sep 24, 2009 #5
    If K(x,u) = u, then what is your operator? Is it linear?
     
  7. Sep 26, 2009 #6

    HallsofIvy

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    It's not linear because you have a function of the dependent variable, u, K(x,u), multiplying the derivative of u, [itex]\partial u/\partial x[/itex].

    (But why are you using the partial derivative if u is a function only of x? Is there another term in the equation with another independent variable?)
     
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