12. Consider the following initial-boundary value problem: Utt = Uxx ux(0, t)u(0, t) = 0, ux(l, t) + u(l, t) = 0. u(x,0) = 0, u(x,0) = 0, x € (0,1), t€ R, ter, x = [0,1]. (i) The above initial-boundary value problem models a vibrating string with two ends attached to elastic springs. The total energy of the whole system (including the vibrating string and two elastic springs) can be expressed as Ẽ(t) - 1/ // (u² + 1² ) dx + 1/ 1²(1.1) + 2/2u² (0,t1). /u²(1, t) Prove that the total energy of the whole system is preserved in time, that is, Ē' (t) = 0.
12. Consider the following initial-boundary value problem: Utt = Uxx ux(0, t)u(0, t) = 0, ux(l, t) + u(l, t) = 0. u(x,0) = 0, u(x,0) = 0, x € (0,1), t€ R, ter, x = [0,1]. (i) The above initial-boundary value problem models a vibrating string with two ends attached to elastic springs. The total energy of the whole system (including the vibrating string and two elastic springs) can be expressed as Ẽ(t) - 1/ // (u² + 1² ) dx + 1/ 1²(1.1) + 2/2u² (0,t1). /u²(1, t) Prove that the total energy of the whole system is preserved in time, that is, Ē' (t) = 0.
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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![12.
Consider the following initial-boundary value problem:
Utt = Uxx,
ux(0, t)u(0, t) = 0, ux(l, t) + u(l, t) = 0.
u(x,0) = 0, ut(x,0) = 0,
x € (0,1), te R,
ter,
x = [0,1].
(i)
The above initial-boundary value problem models a vibrating string with two ends
attached to elastic springs. The total energy of the whole system (including the vibrating string and
two elastic springs) can be expressed as
1
Ẽ(t) = ½ √ √ (u² + u²³) dx +
(u²+u²) dx + 1⁄u²(1,t) + 1⁄2u²(0, t).
Prove that the total energy of the whole system is preserved in time, that is, Ē' (t) = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F75fda764-8052-4cfe-99a9-31213cb8b0c8%2Fb5c109ce-af31-49e0-92d9-61fc23d7d6e6%2Fy5b6v3o_processed.png&w=3840&q=75)
Transcribed Image Text:12.
Consider the following initial-boundary value problem:
Utt = Uxx,
ux(0, t)u(0, t) = 0, ux(l, t) + u(l, t) = 0.
u(x,0) = 0, ut(x,0) = 0,
x € (0,1), te R,
ter,
x = [0,1].
(i)
The above initial-boundary value problem models a vibrating string with two ends
attached to elastic springs. The total energy of the whole system (including the vibrating string and
two elastic springs) can be expressed as
1
Ẽ(t) = ½ √ √ (u² + u²³) dx +
(u²+u²) dx + 1⁄u²(1,t) + 1⁄2u²(0, t).
Prove that the total energy of the whole system is preserved in time, that is, Ē' (t) = 0.
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