(a) Prove that for any C2 solution of this equation, the energy function E(t) = 1 u²dx %3D is a non-increasing function of t. (b) Prove that the following initial value problem и, — (f (х,1)иҳ)ҳ 3g (х,t), 0<х <1,1>0 u(0, t) = u(1,t) = 0, t> 0 u(x,0) = ø(x), 0
(a) Prove that for any C2 solution of this equation, the energy function E(t) = 1 u²dx %3D is a non-increasing function of t. (b) Prove that the following initial value problem и, — (f (х,1)иҳ)ҳ 3g (х,t), 0<х <1,1>0 u(0, t) = u(1,t) = 0, t> 0 u(x,0) = ø(x), 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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PLease answer a and b
Thank you!
Transcribed Image Text:Question 3. Let l > 0, and f (x, t) > 0 a non-negative function on R?. Consider the following
equation with Dirichlet boundary condition:
u; - (f(x,t)ux)x = 0, 0<x < l, t>0
%3D
(1)
и (0, t) %—D и(1,t) 3D 0, 1>0
t >
1
Transcribed Image Text:(a)
Prove that for any C² solution of this equation, the energy function
E() = [' r'dx
1
is a non-increasing function of t.
(b)
Prove that the following initial value problem
u; – (f(x,t)ux), = g(x,t), 0 <x < I, t > 0
и(0,г) %3D и(1,1) %3 0,
t > 0
и(х,0) — ф(х), 0<x<1,
has at most one C² solution. (You can assume the conclusion of part (a) is already proven).
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