Consider the diffusion equation ut kuza for k > 0 on [-1,1] with Robin %3D boundary conditions Uz(-1,t) – au(-1, t) = 0 = u#(1, t)+ au(1,t). (a) If a > 0 show that the energy E(t) = S', [u(x, t)J² dx is decreasing.
Consider the diffusion equation ut kuza for k > 0 on [-1,1] with Robin %3D boundary conditions Uz(-1,t) – au(-1, t) = 0 = u#(1, t)+ au(1,t). (a) If a > 0 show that the energy E(t) = S', [u(x, t)J² dx is decreasing.
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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![H2 Consider the diffusion equation ut =
kua for k > 0 on [-1,1] with Robin
boundary conditions
Uz(-1,t) – au(-1, t) = 0 = u#(1, t) + au(1, t).
(a) If a > 0 show that the energy E(t) = S', [u(x, t)]² dx is decreasing.
(b) If a is much smaller than 0, show by example that energy may increase
or decrease. Hint: Consider solutions of the form h(t) cosh(bx).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a0756d2-342c-4271-9a34-97295939e0e2%2F7ad326a2-6e00-4292-985a-7e49b19fe5fd%2Fnewemtn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:H2 Consider the diffusion equation ut =
kua for k > 0 on [-1,1] with Robin
boundary conditions
Uz(-1,t) – au(-1, t) = 0 = u#(1, t) + au(1, t).
(a) If a > 0 show that the energy E(t) = S', [u(x, t)]² dx is decreasing.
(b) If a is much smaller than 0, show by example that energy may increase
or decrease. Hint: Consider solutions of the form h(t) cosh(bx).
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