"(x) = Af(xi-1) + Bf(x,) + Cf(x\»1) + Df(x\»z) + Ef(x{+x) , 04n In other words, the second derivative of fx) at x = x, must be represented by the combination of the values of fx) at x, itself, one grid point to its left, and the neighboring three grid points to its right, as illustrated in Fig. 2. Show your procedure. Please clearly state the values of A, B, C, D, and E. h f"(x,) +3

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question

Derive a five-point, non-centered, finite difference formula for the second derivative of f(x) at x = xi
that has a truncation error of O(h3). Moreover, the formula must have the following form: 

?′′(??) =((??(??−1) + ??(??) + ??(??+1) + ??(??+2) + ??(??+3))/ℎ2) + ?(ℎ3) .


In other words, the second derivative of f(x) at x = xi must be represented by the combination of the
values of f(x) at xi itself, one grid point to its left, and the neighboring three grid points to its right, as
illustrated in Fig. 2. Show your procedure. Please clearly state the values of A, B, C, D, and E.

c"(x) = Af(x-1) + Bf (x;) + Cf(xi+1) + Df(xi+2) + Ef (xi+3)
h?
+ O(h³).
In other words, the second derivative of f(x) at x = x, must be represented by the combination of the
values of f(x) at x, itself, one grid point to its left, and the neighboring three grid points to its right, as
illustrated in Fig. 2. Show your procedure. Please clearly state the values of A, B, C, D, and E.
f "(x,)
i+1
+2
Transcribed Image Text:c"(x) = Af(x-1) + Bf (x;) + Cf(xi+1) + Df(xi+2) + Ef (xi+3) h? + O(h³). In other words, the second derivative of f(x) at x = x, must be represented by the combination of the values of f(x) at x, itself, one grid point to its left, and the neighboring three grid points to its right, as illustrated in Fig. 2. Show your procedure. Please clearly state the values of A, B, C, D, and E. f "(x,) i+1 +2
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