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Mr. Arfken again

  1. Dec 30, 2008 #1
    1. The problem statement, all variables and given/known data
    Exercise 5.6.24 from Arfken's Essential Mathematical Methods for Physicists. You have a function tabulated at equally spaced values of the argument:
    [tex]\left\{ \begin{array}{c}
    y_n=y(x_n)\\x_n = x+nh \end{array}\right.[/tex]
    Show that the linear combination
    [tex]\frac{1}{12h}\left\{ -y_2+8y_1-8y_{-1}+y_{-2} \right\}[/tex]
    yields
    [tex] y'_0-\frac{h^4}{30}y^{(5)}_0+\cdots . [/tex]
    2. Relevant equations

    I used:
    [tex]\frac{d^ny_0}{dx^n}=\sum_{m=0}^n(-1)^n\left( \begin{array}{c}
    n\\ m \end{array} \right)y_{\frac{n}{2}-m}[/tex] for the derivatives of an even order and a geometric average between the backward and forward derivatives for the odd ones.
    Keep in mind that [tex]y(x)=y_0[/tex], [tex]y(x+h)=y_1[/tex],...
    Also used the Taylor expansion of y(x+h):
    [tex]f(x+h)=\sum_{m=0}^\infty \frac{h^n}{n!}f^{(n)}(x)[/tex].

    3. The attempt at a solution

    I solved the Taylor expansion for [tex] y'_0 [/tex] and tried to modify the result accordingly with the equation the author expressed. But there was no way to get it fit. I guess I didn't understand how the author calculates the derivative of the function.
    Please, someone help. I am getting crazy with the exercises of this book.
     
  2. jcsd
  3. Jan 3, 2009 #2
    Well, I finally got the answer and it was very simple in fact!!
    It's only necessary to expand all the terms enclosed in the parenthesis and truncate the expansion in 5th order.
     
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