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Difference between Essential a Natural BC's

  1. Aug 25, 2012 #1
    Folks,

    I am looking at the Ritz method for the following problem

    ##\displaystyle -\frac{d^2 u}{dx^2}-u+x^2=0## for ##0<x<1##

    with boundary conditions ##u(0)=0## and ##\displaystyle \frac{du}{dx} |_{x=1} =1##

    The last derivative term, how do I know whether that is a natural or essential BC?

    I have googled the following guidelines but I am still confused.

    Specification of the primary variable ( u in this case) is an essential BC*
    Specification of a secondary variable (like a force F, not present in this example) is a natural boundary condition

    IF a boundary condition involves one or more variables in a 'direct' way it is essential otherwise it is natural.
    Direct implies excluding derivative of the primary function.**

    I find this info conflicting based on * and **
    I think the book states it is a natural BC.

    Would appreciate some clarification...
    THanks
     
  2. jcsd
  3. Aug 25, 2012 #2
    As I understand the difference:

    What is meant is that direct gives an expression that yields a definite value for (in this case) u.

    for example u(0) = 0 says that at x=0 the value of u is zero.

    This is contrasted by natural expression which does not lead to a definite value of u.

    for example


    [tex]{\left[ {\frac{{du}}{{dx}}} \right]_{x = 1}} = 1[/tex]

    does not yield a definite value for u at x = 1 since a curve of slope 1 can be drawn through any value of u.

    However this is really just classification for the sake of it and nothing to worry about.
     
  4. Aug 27, 2012 #3
    Thank you sir, that explains it nicely. I might be back with other BC type q's :-)
     
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