Physics Help Forum Taylor derivation of three-point forward difference formula

 Feb 27th 2013, 04:21 PM #1 Member   Join Date: Feb 2013 Location: Greater St. Louis area Posts: 43 Taylor derivation of three-point forward difference formula I have a test early next week on using the Taylor series. Does the attached derivation look correct and is the error term correct? Thank you all for your help so far, especially Dr. Phil. Last edited by roger; Mar 31st 2015 at 01:55 PM.
Feb 28th 2013, 10:57 AM   #2
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 Originally Posted by roger I have a test early next week on using the Taylor series. Does the attached derivation look correct and is the error term correct? Thank you all for your help so far.
What you have done (correctly) is to verify the given equation - it might be fun to try to derive it from scratch. I made a table with 1st and 2nd differences, including backward extrapolation to (t-tau)
Code:

t-tau |Extrapolate   |                  |                          |
|              |Extrapolate       |                          |
t     |f(t)          |                  |f(t+2tau)-2f(t+tau)+f(tau)|
|              |f(t+tau)-f(t)     |                          |Error Term
t+tau |f(t)+..+..+.. |                  |f(t+2tau)-2f(t+tau)+f(tau)|
|              |f(t+2tau)-f(t-tau)|                          |
t+2tau|f(t)+2..+4..+8|                  |                          |
Since we are neglecting the 3rd difference, it is the Error Term. Thus the 2nd difference is assumed to be constant, allowing you to calculate the 1st difference between t and t-tau. Then set

f'(t) = [sum of 1st differences]/(2 tau) = [f(t+tau) - f(t-tau)]/(2 tau)

That should produce the desired formula.

Looking at the "explicit expression" for the error term -- when you take 4(A) - (B), isn't it -(2/3)tau^3 f'''(t) ? So the final expression would be (1/3)tau^2)|f'''(t)| ?

If you try to complete the derivation as outlined above, you can also extrapolate the error term back to see how it affects f'(t).

[BTW - for math help you can also try the forums at www.FreeMathHelp.com - they have a working LaTeX over there that makes it easier to "type" math.]
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Last edited by DrPhil; Feb 28th 2013 at 11:19 AM. Reason: sign error in 2nd diff.

 Tags derivation, difference, formula, forward, taylor, threepoint

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