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Summation Notation for Weak form of Differential Equation

  1. Sep 8, 2012 #1
    Folks,

    I am struggling to see what is happening here particularly when ## \displaystyle \sum_{i=1}^{n-1}## transforms into ##\displaystyle \int_{x_1^e}^{x_{n}^e}##

    ##\displaystyle 0=\sum_{i=1}^{n-1} \left [ \int_{x_i^e}^{x_{i+1}^e} (a \frac{dw}{dx} \frac{du}{dx}+cwu-wf )dx- \left [ w(x) a \frac{du}{dx} \right ]_{x_i^e}^{x_{i+1}^e} \right ]##

    ##\displaystyle =\int_{x_1^e}^{x_{n}^e} (a \frac{dw}{dx} \frac{du}{dx}+cwu-wf )dx -w(x_1^e) (-a \frac{du}{dx})_{x_1^e}-w(x_2^e) (a \frac{du}{dx})_{x_2^e}-w(x_2^e) (-a \frac{du}{dx})_{x_2^e}-w(x_3^e) (a \frac{du}{dx})_{x_3^e} ......-w(x_{n-1}^e) (-a \frac{du}{dx})_{x_{n-1}^e}-w(x_{n}^e) (a \frac{du}{dx})_{x_{n}^e}##

    Could anyone shed some light on what is happening..?

    Apparently it is carried out in intervals of ##(x_1^e, x_2^e), (x_2^e, x_3^e),(x_{n-1}^e, x_n^e)##

    Thanks
     
  2. jcsd
  3. Sep 12, 2012 #2
    I am no expert on this, but I think hes using a the property of integration
    [tex] \int^{b}_{a} f + \int^{c}_{b} f = \int^{c}_{a} f [/tex]
    the intermediate terms are just writing out the [tex][w(x)a\frac{du}{dx}]^{x^{e}_{i+1}}_{x^{e}_{i}} [/tex] explicitly
     
  4. Sep 14, 2012 #3
    Thanks. That helps my interpretation. I knew the intermediate terms.
    Regards
     
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