3. Use Taylor series expansions to determine the error in the approxima- u(t+3h)-3u(t+2h)+3u(t+h)-u(t) h3 tion u"(t) 2 uG+3h)=u+3hu't(36j2 ull + (34Ý u"'s (34)" %3D 24 -3 uk+26) =u+2hu' +(262 u" f of to u (fth) =uthul a u" (36) num = 24 %3D Dind 4 R u.hioh meke +he opproximotin
3. Use Taylor series expansions to determine the error in the approxima- u(t+3h)-3u(t+2h)+3u(t+h)-u(t) h3 tion u"(t) 2 uG+3h)=u+3hu't(36j2 ull + (34Ý u"'s (34)" %3D 24 -3 uk+26) =u+2hu' +(262 u" f of to u (fth) =uthul a u" (36) num = 24 %3D Dind 4 R u.hioh meke +he opproximotin
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
100%
Answer is given BUT need full detailed steps and
process since I don't understand the concept.
![3. Use Taylor series expansions to determine the error in the approximation \( u'''(t) \approx \frac{u(t+3h) - 3u(t+2h) + 3u(t+h) - u(t)}{h^3} \).
1. \( u(t+3h) = u + 3h u' + \frac{(3h)^2}{2} u'' + \frac{(3h)^3}{6} u''' + \frac{(3h)^4}{24} u^{(4)} + \ldots \)
\(-3 \quad u(t+2h) = u + 2h u' + \frac{(2h)^2}{2} u'' + \frac{(2h)^3}{6} u''' + \frac{(2h)^4}{24} u^{(4)} + \ldots \)
3. \( u(t+h) = u + h u' + \frac{h^2}{2} u'' + \frac{h^3}{6} u''' + \frac{h^4}{24} u^{(4)} + \ldots \)
\(-1 \quad u(t) = u \)
---
Expanding the numerators:
\[ \text{num} = 0 \quad 0 \quad 0 \]
\[ \frac{\text{num}}{h^3} = u'''(t) + \frac{3}{2} h u^{(4)} \]
\[ \frac{h^3}{6} u^{(3)}(6) + \frac{h^4}{24} u^{(4)}(36) + \ldots \]
This is the transcription and breakdown of the provided handwritten text. It shows the use of Taylor series to analyze the approximation error of the third derivative of a function \(u(t)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2Ff79f8be4-4a7a-4617-ae41-635d2b485ab3%2F1zmfgwq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Use Taylor series expansions to determine the error in the approximation \( u'''(t) \approx \frac{u(t+3h) - 3u(t+2h) + 3u(t+h) - u(t)}{h^3} \).
1. \( u(t+3h) = u + 3h u' + \frac{(3h)^2}{2} u'' + \frac{(3h)^3}{6} u''' + \frac{(3h)^4}{24} u^{(4)} + \ldots \)
\(-3 \quad u(t+2h) = u + 2h u' + \frac{(2h)^2}{2} u'' + \frac{(2h)^3}{6} u''' + \frac{(2h)^4}{24} u^{(4)} + \ldots \)
3. \( u(t+h) = u + h u' + \frac{h^2}{2} u'' + \frac{h^3}{6} u''' + \frac{h^4}{24} u^{(4)} + \ldots \)
\(-1 \quad u(t) = u \)
---
Expanding the numerators:
\[ \text{num} = 0 \quad 0 \quad 0 \]
\[ \frac{\text{num}}{h^3} = u'''(t) + \frac{3}{2} h u^{(4)} \]
\[ \frac{h^3}{6} u^{(3)}(6) + \frac{h^4}{24} u^{(4)}(36) + \ldots \]
This is the transcription and breakdown of the provided handwritten text. It shows the use of Taylor series to analyze the approximation error of the third derivative of a function \(u(t)\).
Expert Solution
Step 1
_1.jpg)
Step by step
Solved in 3 steps with 3 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,
