A cylindrical homogenous, isotropic rod with insulated sides has the temperature f(x) L/2 ={L-X L2SXSL at time t = 0. For time / > 0 its ends (at x = 0 and x = L) are insulated.
A cylindrical homogenous, isotropic rod with insulated sides has the temperature f(x) L/2 ={L-X L2SXSL at time t = 0. For time / > 0 its ends (at x = 0 and x = L) are insulated.
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
1
![A cylindrical homogenous, isotropic
rod with insulated sides has the
temperature
= {
f(x) = {
L-x
0≤x≤ L/2
L/2 ≤ x ≤L
at time t = 0. For time t > 0 its
ends (at x = 0 and x = L) are
insulated.
[a] Find a formula for the
temperature U(x, t) in the rod for
0 < x < L and t > 0.
[b] Find lim,→∞ U(x, t). Interpret
the value of the limit physically.
[c] Find the heat flow rate for the
cross-section of the rod at x = 0
and x = L/2. What is the physical
interpretation of your answer for
x = L/2?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32c3ae95-8712-4682-9e1f-ff7c3e49cdb8%2F662d041b-d1cb-402b-bfec-cd19d68a9af4%2Fa7r9g8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A cylindrical homogenous, isotropic
rod with insulated sides has the
temperature
= {
f(x) = {
L-x
0≤x≤ L/2
L/2 ≤ x ≤L
at time t = 0. For time t > 0 its
ends (at x = 0 and x = L) are
insulated.
[a] Find a formula for the
temperature U(x, t) in the rod for
0 < x < L and t > 0.
[b] Find lim,→∞ U(x, t). Interpret
the value of the limit physically.
[c] Find the heat flow rate for the
cross-section of the rod at x = 0
and x = L/2. What is the physical
interpretation of your answer for
x = L/2?
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