Consider the heat equationUt = AUxx, a €R>⁰, 00with boundary conditionsux (0, t) = ux (L, t) = 0, t> 0.i. Give a meaningful interpretation of the boundary conditions.ii. Reconsider the problem in part a. of this question, which concerns the temperaturein a one dimensional carbon composite bar of length 1000 millimeters, but nowlet it have the boundary conditions in part b. Find a solution, u(x, t), for thetemperature in the bar. You will need to show your method, which will includeconsidering all eigenvalues of the problem. Your answer will contain arbitraryconstants (because we have not specified an initial condition).iii. What will happen to the temperature in the bar after a very long time?iv. If the bar were made of iron, which has a thermal diffusivity of approximately20mm²/s, would the heat flow in the bar change in any way? If so, how?

i) In the heat equation the unknown variable is temparature. i,e the heat flows into (respectively,…

Answered: Consider the heat equation Ut = AUxx, a…

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Advanced Math

Consider the heat equation Ut = AUxx, a €R>⁰, 00 with boundary conditions ux (0, t) = ux (L, t) = 0, 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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Question

Consider the heat equation
Ut = AUxx, a €R>⁰, 0<x<L, t>0
with boundary conditions
ux (0, t) = ux (L, t) = 0, t> 0.
i. Give a meaningful interpretation of the boundary conditions.
ii. Reconsider the problem in part a. of this question, which concerns the temperature
in a one dimensional carbon composite bar of length 1000 millimeters, but now
let it have the boundary conditions in part b. Find a solution, u(x, t), for the
temperature in the bar. You will need to show your method, which will include
considering all eigenvalues of the problem. Your answer will contain arbitrary
constants (because we have not specified an initial condition).
iii. What will happen to the temperature in the bar after a very long time?
iv. If the bar were made of iron, which has a thermal diffusivity of approximately
20mm²/s, would the heat flow in the bar change in any way? If so, how?

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