utt - (rux + X(a,b) Utx)x - Byt = 0, x €]0, 1[, t > 0, Ytt - (Yx + X(a,b) Ytx)x+ But = 0, x €]0, 1[, t > 0, u(0, t) = u(1, t) = y(0, t) = y(1, t) = 0, t > 0, u(x,0) = u(x), y(x,0) = yo(x), x €]0, 1[ ut(x, 0) = u₁(x), yt(x,0) = y₁(x), x €]0, 1[, where r, 3 are positive constants and 0 < a < b < 1 1. Find the energy E(t) of the system. 2. Define the appropriate Hilbert energy space H and its inner product. 3. Find the linear unbounded operator A and its domain D(A), and write the system in the form = AU Ut U(0) = Uo € H 4. Show that A is m-dissipative (for maximality: prove that 0 € p(A)). (1) 5. Deduce the existence and uniqueness of system (1). 6. Prove that the energy of this system tends to zero as t goes to infinity (non-compact resolvent). 7. Replace the the Kelvin-voigt dampings by frictional dampings (compact resolvent), then prove that the energy of this system tends to zero as t goes to infinity.
utt - (rux + X(a,b) Utx)x - Byt = 0, x €]0, 1[, t > 0, Ytt - (Yx + X(a,b) Ytx)x+ But = 0, x €]0, 1[, t > 0, u(0, t) = u(1, t) = y(0, t) = y(1, t) = 0, t > 0, u(x,0) = u(x), y(x,0) = yo(x), x €]0, 1[ ut(x, 0) = u₁(x), yt(x,0) = y₁(x), x €]0, 1[, where r, 3 are positive constants and 0 < a < b < 1 1. Find the energy E(t) of the system. 2. Define the appropriate Hilbert energy space H and its inner product. 3. Find the linear unbounded operator A and its domain D(A), and write the system in the form = AU Ut U(0) = Uo € H 4. Show that A is m-dissipative (for maximality: prove that 0 € p(A)). (1) 5. Deduce the existence and uniqueness of system (1). 6. Prove that the energy of this system tends to zero as t goes to infinity (non-compact resolvent). 7. Replace the the Kelvin-voigt dampings by frictional dampings (compact resolvent), then prove that the energy of this system tends to zero as t goes to infinity.
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
Number 7 please
![Ytt -
Utt- - (rux + X(a,b) Utx)x - Byt = 0, x ]0, 1[, t > 0,
(Yx + X(a,b) Ytx) x + But = 0, x ≤]0, 1[, t > 0,
u(0, t) = u(1, t) = y(0, t) = y(1, t) = 0, t > 0,
u(x,0) = u₁(x), y(x,0) = yo(x), x ≤]0, 1[
ut (x, 0) = u₁(x), yt(x,0) = y₁(x), x ≤]0, 1[,
where r, 3 are positive constants and 0 < a < b < 1
1. Find the energy E(t) of the system.
2. Define the appropriate Hilbert energy space H and its inner product.
3. Find the linear unbounded operator A and its domain D(A), and write the system
in the form
Ut = AU
\ U(0) =Uo ∈ H
4. Show that A is m-dissipative (for maximality: prove that 0 € p(A)).
5. Deduce the existence and uniqueness of system (1).
(1)
6. Prove that the energy of this system tends to zero as t goes to infinity (non-compact
resolvent).
7. Replace the the Kelvin-voigt dampings by frictional dampings (compact resolvent),
then prove that the energy of this system tends to zero as t goes to infinity.
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9612a6a5-e19c-4346-ae6e-6941d4a9f0e7%2F79d038ed-d298-4572-ae82-8bd495f18f39%2F9tbgxwm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Ytt -
Utt- - (rux + X(a,b) Utx)x - Byt = 0, x ]0, 1[, t > 0,
(Yx + X(a,b) Ytx) x + But = 0, x ≤]0, 1[, t > 0,
u(0, t) = u(1, t) = y(0, t) = y(1, t) = 0, t > 0,
u(x,0) = u₁(x), y(x,0) = yo(x), x ≤]0, 1[
ut (x, 0) = u₁(x), yt(x,0) = y₁(x), x ≤]0, 1[,
where r, 3 are positive constants and 0 < a < b < 1
1. Find the energy E(t) of the system.
2. Define the appropriate Hilbert energy space H and its inner product.
3. Find the linear unbounded operator A and its domain D(A), and write the system
in the form
Ut = AU
\ U(0) =Uo ∈ H
4. Show that A is m-dissipative (for maximality: prove that 0 € p(A)).
5. Deduce the existence and uniqueness of system (1).
(1)
6. Prove that the energy of this system tends to zero as t goes to infinity (non-compact
resolvent).
7. Replace the the Kelvin-voigt dampings by frictional dampings (compact resolvent),
then prove that the energy of this system tends to zero as t goes to infinity.
1
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 4 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,
