All of the formulas in Table 8-1 in the textbook have a truncation error of O(h), O(h2), or O(h4). Try to derive a four-point, non-centered, finite difference formula for the first derivative of f(x) at x = xi that has a truncation error of O(h3). Moreover, the formula must have the following form: ?′(??) =((??(??−2) + ??(??−1) + ??(??) + ??(??+1))/ℎ)+ ?(ℎ3) In other words, the derivative of f(x) at x = xi must be represented by the combination of the values of f(x) at xi itself, one neighboring grid point to its right, and two grid points to its left, as illustrated in Fig. 1. Show your procedure. Please clearly state the values of A, B, C, and D.
All of the formulas in Table 8-1 in the textbook have a truncation error of O(h), O(h2), or O(h4). Try to derive a four-point, non-centered, finite difference formula for the first derivative of f(x) at x = xi that has a truncation error of O(h3). Moreover, the formula must have the following form: ?′(??) =((??(??−2) + ??(??−1) + ??(??) + ??(??+1))/ℎ)+ ?(ℎ3) In other words, the derivative of f(x) at x = xi must be represented by the combination of the values of f(x) at xi itself, one neighboring grid point to its right, and two grid points to its left, as illustrated in Fig. 1. Show your procedure. Please clearly state the values of A, B, C, and D.
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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All of the formulas in Table 8-1 in the textbook have a truncation error of O(h), O(h2), or O(h4). Try to derive a four-point, non-centered, finite difference formula for the first derivative of f(x) at x = xi that has a truncation error of O(h3). Moreover, the formula must have the following form:
?′(??) =((??(??−2) + ??(??−1) + ??(??) + ??(??+1))/ℎ)+ ?(ℎ3)
In other words, the derivative of f(x) at x = xi must be represented by the combination of the values of f(x) at xi itself, one neighboring grid point to its right, and two grid points to its left, as illustrated in Fig. 1. Show your procedure. Please clearly state the values of A, B, C, and D.
Transcribed Image Text:L'x) = Af (x1-2) + Bf(x¡-1) + Cf(x;) + Df(xias), o4
h
In other words, the derivative of fx) at x =x, must be represented by the combination of the values of flx)
at x, itself, one neighboring grid point to its right, and two grid points to its left, as illustrated in Fig. 1.
Show your procedure. Please clearly state the values of A, B, C, and D.
h
f (x)
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