2. Define the appropriate Hilbert energy space H and its inner product.
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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Question
Number 2 please
Transcribed Image Text:− (u² + r²u² + 2X (a,b) UzUtx + ß²y² + y² + 2X(a,b) Yx Ytx)
The energy E(t) of the system is obtained by integrating the energy density over the spatial domain:
E(t) =
E(x, t) dx
E(x, t)
=
![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[
u₁(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.
(1)
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9612a6a5-e19c-4346-ae6e-6941d4a9f0e7%2Facf283ae-3901-41ae-9cc1-72cb76d60f87%2Fbcehsgo_processed.jpeg&w=3840&q=75)
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[
u₁(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.
(1)
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
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