- Consider an eigenvalue problem -u" - Au = 0; uES = C₂ (0,00) and VvE S, v'(0) = 0 & v(co) = ok. (a) Determine the Green's function G(x,; 2)for the problem. (b) Derive the spectral representation by utilizing the identity. 1 R→∞ 2πi 8(x - 5) = - lim (c) Obtain the Fourier cosine transform pair & G(x, 5; 2) da CR 8 f(x) = f f(x) coskx dk -8
- Consider an eigenvalue problem -u" - Au = 0; uES = C₂ (0,00) and VvE S, v'(0) = 0 & v(co) = ok. (a) Determine the Green's function G(x,; 2)for the problem. (b) Derive the spectral representation by utilizing the identity. 1 R→∞ 2πi 8(x - 5) = - lim (c) Obtain the Fourier cosine transform pair & G(x, 5; 2) da CR 8 f(x) = f f(x) coskx dk -8
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
![4. Consider an eigenvalue problem
-u" - Au = 0; u ES = C₂ (0, ∞o) and VvE S, v'(0) = 0 & v(0) = ok.
(a) Determine the Green's function G(x, 5; 2)for the problem.
(b) Derive the spectral representation by utilizing the identity.
1
8(x - 5) = - lim ZTri Per
if
R→∞⁰0 2πi
(c) Obtain the Fourier cosine transform pair
CR
G(x, }; λ)dλ
f(x) = [ f(x) coskx dk](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3902a4e8-914f-4f9b-ac8a-e53660fdf37b%2Fb1587d7f-9672-4482-939f-bdaff72c9df0%2Fmukx9kk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Consider an eigenvalue problem
-u" - Au = 0; u ES = C₂ (0, ∞o) and VvE S, v'(0) = 0 & v(0) = ok.
(a) Determine the Green's function G(x, 5; 2)for the problem.
(b) Derive the spectral representation by utilizing the identity.
1
8(x - 5) = - lim ZTri Per
if
R→∞⁰0 2πi
(c) Obtain the Fourier cosine transform pair
CR
G(x, }; λ)dλ
f(x) = [ f(x) coskx dk
![ƒ (k):
=
2
TT
J₁ f(x) coskx dx.
88
(d) Solve the following initial value problem for u(t,x):
² u
əx²
ди
at
Satisfying the boundary and initial conditions
ди
əx
u is ok as x→∞0&t> 0
=
= 1 at x = 0 &t> 0
u= 0 at t=0&x > 0.
Is this solution uniformly convergent?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3902a4e8-914f-4f9b-ac8a-e53660fdf37b%2Fb1587d7f-9672-4482-939f-bdaff72c9df0%2Fmuydpt7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ƒ (k):
=
2
TT
J₁ f(x) coskx dx.
88
(d) Solve the following initial value problem for u(t,x):
² u
əx²
ди
at
Satisfying the boundary and initial conditions
ди
əx
u is ok as x→∞0&t> 0
=
= 1 at x = 0 &t> 0
u= 0 at t=0&x > 0.
Is this solution uniformly convergent?
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
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 5 steps with 6 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Similar questions
- Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,