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 < 11. 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 systemin the formUt = AU\ U(0) =Uo ∈ H4. 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-compactresolvent).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

As per the question, we consider a system governed by a set of partial differential equations with…

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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.3: Algebraic Expressions

Problem 20E

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