Use Taylor series expansions to determine the error in the approximation
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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![**Using Taylor Series Expansions to Determine the Error in the Approximation**
To approximate the fourth derivative of \( u \) at \( t \), we use the following expression:
\[
u^{iv}(t) \approx \frac{u(t+2h) - 4u(t+h) + 6u(t) - 4u(t-h) + u(t-2h)}{h^4}
\]
This simplifies to:
\[
= \frac{h^4 u^{iv} + \frac{1}{6} h^6 u^{(vi)}}{h^4}
\]
\[
= u^{iv} + \frac{1}{6} h^2 u^{(vi)}
\]
The last term, \(\frac{1}{6} h^2 u^{(vi)}\), represents the error in the approximation.
**Hint**: Expand \( u(t + 2h) \), etc., using Taylor series up to the \( h^6 \) term.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2F37098954-cb78-45f1-82d2-a1adb6066165%2Fpgo1mcf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using Taylor Series Expansions to Determine the Error in the Approximation**
To approximate the fourth derivative of \( u \) at \( t \), we use the following expression:
\[
u^{iv}(t) \approx \frac{u(t+2h) - 4u(t+h) + 6u(t) - 4u(t-h) + u(t-2h)}{h^4}
\]
This simplifies to:
\[
= \frac{h^4 u^{iv} + \frac{1}{6} h^6 u^{(vi)}}{h^4}
\]
\[
= u^{iv} + \frac{1}{6} h^2 u^{(vi)}
\]
The last term, \(\frac{1}{6} h^2 u^{(vi)}\), represents the error in the approximation.
**Hint**: Expand \( u(t + 2h) \), etc., using Taylor series up to the \( h^6 \) term.
Expert Solution
Step 1
In this question we expand by using formula
up to h6
Step by step
Solved in 2 steps with 1 images
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