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Problem 13.3.5: Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form a?u k əx? ди · hu = 0 < x < L, t> 0 where h is a constant. Find the temperature u(x,t) if the initial temperature is f (x) throughout and the ends x = 0 and x = L are insulated. See figure below.

Problem 13.3.5: Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form a?u k əx? ди · hu = 0 < x < L, t> 0 where h is a constant. Find the temperature u(x,t) if the initial temperature is f (x) throughout and the ends x = 0 and x = L are insulated. See figure below.

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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Problem 13.3.5: Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form a²u k əx² ди һи — at' 0 < x < L, t > 0 where h is a constant. Find the temperature u(x,t) if the initial temperature is f (x) throughout and the ends x = 0 and x = L are insulated. See figure below. insulated 0° insulated L 0° heat transfer from lateral surface of the rod
Transcribed Image Text:Problem 13.3.5: Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form a²u k əx² ди һи — at' 0 < x < L, t > 0 where h is a constant. Find the temperature u(x,t) if the initial temperature is f (x) throughout and the ends x = 0 and x = L are insulated. See figure below. insulated 0° insulated L 0° heat transfer from lateral surface of the rod
We can expand f (x) into its Fourier series (sine only) E Bu sin ( ) - bn sin (x) Ex) = f(x) - I B, Sin %3D n=1 with 2 (-) bn f (x) sin x ) dx Compare coefficients and find B, = bn and find the same answer as before: Σ -n?n? at L2 2 y(x, t) : L (x)e f(x) sin -x) dx ) sin L n=1
Transcribed Image Text:We can expand f (x) into its Fourier series (sine only) E Bu sin ( ) - bn sin (x) Ex) = f(x) - I B, Sin %3D n=1 with 2 (-) bn f (x) sin x ) dx Compare coefficients and find B, = bn and find the same answer as before: Σ -n?n? at L2 2 y(x, t) : L (x)e f(x) sin -x) dx ) sin L n=1
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