Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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]
    [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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook