(a) Write down the Taylor series for f(x+^x and f(x - Ax) about the point x. Explicitly keep track of terms of order Ax4 and lower. (b) Derive the central-difference O(Ax²) accurate scheme for the second derivative using the Taylor series for f(x± Ax) from above. Be sure to keep the truncation error. (c) Construct the differentiation matrix D which takes the first derivative of a vector f of length n, using the following central-difference scheme: f(x + ▲x) − f (x · 2Ax +0(Ax²) Assume periodic boundary conditions, so that f(n + 1) = f(1). - Ax) f'(x) =
(a) Write down the Taylor series for f(x+^x and f(x - Ax) about the point x. Explicitly keep track of terms of order Ax4 and lower. (b) Derive the central-difference O(Ax²) accurate scheme for the second derivative using the Taylor series for f(x± Ax) from above. Be sure to keep the truncation error. (c) Construct the differentiation matrix D which takes the first derivative of a vector f of length n, using the following central-difference scheme: f(x + ▲x) − f (x · 2Ax +0(Ax²) Assume periodic boundary conditions, so that f(n + 1) = f(1). - Ax) f'(x) =
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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![**Address the following questions:**
(a) Write down the Taylor series for \( f(x+\Delta x) \) and \( f(x-\Delta x) \) about the point \( x \). Explicitly keep track of terms of order \(\Delta x^4\) and lower.
(b) Derive the central-difference \( \mathcal{O}(\Delta x^2) \) accurate scheme for the second derivative using the Taylor series for \( f(x \pm \Delta x) \) from above. Be sure to keep the truncation error.
(c) Construct the differentiation matrix \(\mathbf{D}\) which takes the first derivative of a vector \(\mathbf{f}\) of length \( n \), using the following central-difference scheme:
\[
f'(x) = \frac{f(x+\Delta x) - f(x-\Delta x)}{2\Delta x} + \mathcal{O}(\Delta x^2)
\]
Assume *periodic* boundary conditions, so that \( f(n+1) = f(1) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6e2f997-9120-4975-9388-a1bc7e4c3a16%2Fd434823a-4195-4bdc-9f22-2f1e63d123a5%2Fo37irgi_processed.png&w=3840&q=75)
Transcribed Image Text:**Address the following questions:**
(a) Write down the Taylor series for \( f(x+\Delta x) \) and \( f(x-\Delta x) \) about the point \( x \). Explicitly keep track of terms of order \(\Delta x^4\) and lower.
(b) Derive the central-difference \( \mathcal{O}(\Delta x^2) \) accurate scheme for the second derivative using the Taylor series for \( f(x \pm \Delta x) \) from above. Be sure to keep the truncation error.
(c) Construct the differentiation matrix \(\mathbf{D}\) which takes the first derivative of a vector \(\mathbf{f}\) of length \( n \), using the following central-difference scheme:
\[
f'(x) = \frac{f(x+\Delta x) - f(x-\Delta x)}{2\Delta x} + \mathcal{O}(\Delta x^2)
\]
Assume *periodic* boundary conditions, so that \( f(n+1) = f(1) \).
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