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The temperature u(x,t) in a bar of silver of length L = 10cm, density p = 10.6-, thermal

Numerical Analysis | 10th Edition | ISBN: 9781305253667 | Authors: Richard L. Burden J. Douglas Faires, Annette M. Burden ISBN: 9781305253667 457

Solution for problem 14 Chapter 8.5

Numerical Analysis | 10th Edition

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Numerical Analysis | 10th Edition | ISBN: 9781305253667 | Authors: Richard L. Burden J. Douglas Faires, Annette M. Burden

Numerical Analysis | 10th Edition

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Problem 14

The temperature u(x,t) in a bar of silver of length L = 10cm, density p = 10.6-, thermal cal cal cm conductivity K 1.04 , and specific heater = 0.056 that is laterally insulated cm * deg * s gm * deg and whose ends are kept at 0 oC is governed by the heat equation d 8 2 u(x, t) B m(x, r), 0 < x < L, 0 < r dt dxwith boundary conditions m(0, r) = 0 and u{L,t) 0 and initial condition m(x,0) = /(x) = lOx x 2 . The solution to the problem is given by 00 , "(x, 0 = ^a exp ( =i ^ P 2 n 2 n 2 \ . fnn -t sin x where fi = ^ and the coefficients are from the Fouriersine series a i< s ' n {t'x ) f or f(x ) where a " = I Jo /Wsin {tx ) cIx - a. Find the first four nonzero terms ofthe Fourier sine series of f(x) = lOx x 2 . b. Compare f(x) to the first four nonzero terms of a\ sin (^i) + 02 sin (2^) + cit, sin (3^) + fla sin (4^) + for x = 3, 6. 9. c. Find the first four nonzero terms of u{x,t). d. Approximate m(9, 0.5), t((6, 0.75), m(3, 1) using part (c).

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7/26/2017 OneNote Online 3.5 Tuesday, October 7, 29:33 AM https://onedrive.live.com/view.aspxref=button&Bsrc=SMIT&resid=36773184373A8F0B!2562&cid=36773184373a8f0b&app=OneNote&authkey=Avz_e_hLmB4xJLw 1/7 7/26/2017 OneNote Online https://onedrive.live.com/view.aspxref=button&Bsrc=SMIT&resid=36773184373A8F0B!2562&cid=36773184373a8f0b&app=OneNote&authkey=Avz_e_hLmB4xJLw 2/7 7/26/2017

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Chapter 8.5, Problem 14 is Solved
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Textbook: Numerical Analysis
Edition: 10
Author: Richard L. Burden J. Douglas Faires, Annette M. Burden
ISBN: 9781305253667

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The temperature u(x,t) in a bar of silver of length L = 10cm, density p = 10.6-, thermal