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

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
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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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?
Transcribed Image Text: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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