2. Use Taylor series expansions to determine the error in the approxima- tion u'(x) = 3u(a)-4u(x-h)+u(x-2h) 2h 3 Uk)=4 ー ucef)= u-hu' t h'u" £ fri emor numendr = 2hur h'u"+- %3D tro = u&-ズ") + そ。。
2. Use Taylor series expansions to determine the error in the approxima- tion u'(x) = 3u(a)-4u(x-h)+u(x-2h) 2h 3 Uk)=4 ー ucef)= u-hu' t h'u" £ fri emor numendr = 2hur h'u"+- %3D tro = u&-ズ") + そ。。
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Answer is given BUT need full detailed steps and
process since I don't understand the concept.
![**Title: Understanding Taylor Series Expansions for Error Approximation**
**Objective:**
Determine the error in the approximation \( u'(x) \approx \frac{3u(x) - 4u(x-h) + u(x-2h)}{2h} \) using Taylor series expansions.
**Content:**
1. **Introduction:**
The goal is to evaluate the accuracy of the numerical differentiation formula given by the expression above. This involves using Taylor series expansions to understand the error term in the approximation of the first derivative \( u'(x) \).
2. **Taylor Series Expansions:**
- **Taylor expansion for \( u(x-h) \):**
\[
u(x-h) = u - hu' + \frac{h^2}{2}u'' - \frac{h^3}{6}u''' + \ldots
\]
- **Taylor expansion for \( u(x-2h) \):**
\[
u(x-2h) = u - 2hu' + \frac{4h^2}{2}u'' - \frac{8h^3}{6}u''' + \ldots
\]
3. **Approximation Analysis:**
- Starting with \( 3u(x) = 3u \).
- Using expansions to substitute for \( u(x-h) \) and \( u(x-2h) \).
- Combine and simplify:
- Numerator:
\[
\text{numerator} = 2hu' - \frac{4h^3}{6}u''' + \ldots
\]
- Divide by \( 2h \):
\[
\frac{2hu' - \frac{4h^3}{6}u''' + \ldots}{2h} = u'(x) - \frac{h^2}{3}u'''(x) + \ldots
\]
4. **Error Term:**
- The error in the approximation is the term \( -\frac{h^2}{3}u'''(x) + \ldots \).
- This highlights that the leading error term in the Taylor expansion is proportional to \( h^2 \), which indicates that the approximation's accuracy improves with smaller values of \( h \).
**Conclusion:**
The analysis](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2Fb277f76d-5881-4542-9fea-dcacb33a2a1b%2Feu2b3be_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Taylor Series Expansions for Error Approximation**
**Objective:**
Determine the error in the approximation \( u'(x) \approx \frac{3u(x) - 4u(x-h) + u(x-2h)}{2h} \) using Taylor series expansions.
**Content:**
1. **Introduction:**
The goal is to evaluate the accuracy of the numerical differentiation formula given by the expression above. This involves using Taylor series expansions to understand the error term in the approximation of the first derivative \( u'(x) \).
2. **Taylor Series Expansions:**
- **Taylor expansion for \( u(x-h) \):**
\[
u(x-h) = u - hu' + \frac{h^2}{2}u'' - \frac{h^3}{6}u''' + \ldots
\]
- **Taylor expansion for \( u(x-2h) \):**
\[
u(x-2h) = u - 2hu' + \frac{4h^2}{2}u'' - \frac{8h^3}{6}u''' + \ldots
\]
3. **Approximation Analysis:**
- Starting with \( 3u(x) = 3u \).
- Using expansions to substitute for \( u(x-h) \) and \( u(x-2h) \).
- Combine and simplify:
- Numerator:
\[
\text{numerator} = 2hu' - \frac{4h^3}{6}u''' + \ldots
\]
- Divide by \( 2h \):
\[
\frac{2hu' - \frac{4h^3}{6}u''' + \ldots}{2h} = u'(x) - \frac{h^2}{3}u'''(x) + \ldots
\]
4. **Error Term:**
- The error in the approximation is the term \( -\frac{h^2}{3}u'''(x) + \ldots \).
- This highlights that the leading error term in the Taylor expansion is proportional to \( h^2 \), which indicates that the approximation's accuracy improves with smaller values of \( h \).
**Conclusion:**
The analysis
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
