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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 U₁ = AU U (0) =Uo € H

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 U₁ = AU U (0) =Uo € H

Advanced Engineering Mathematics
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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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, ß 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 U₁ = AU \ 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. 1
Transcribed Image Text: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, ß 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 U₁ = AU \ 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. 1
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