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Consider the heat distribution u (x, t) in a semi-infinite rod. Assume that the finite end is kept in contact with ice at 0°C and if initially the rod is submerged in hot water at 100°C. (a) Use the appropriate Fourier transform to show that (explain the choice of the Fourier transform) d û (a, t) + c²a²û (a, t) = 100ac² with Au (a,0) = 0, dt 70 where is the appropriate Fourier transform of u and c is the specific heat of the rod.

Consider the heat distribution u (x, t) in a semi-infinite rod. Assume that the finite end is kept in contact with ice at 0°C and if initially the rod is submerged in hot water at 100°C. (a) Use the appropriate Fourier transform to show that (explain the choice of the Fourier transform) d û (a, t) + c²a²û (a, t) = 100ac² with Au (a,0) = 0, dt 70 where is the appropriate Fourier transform of u and c is the specific heat of the rod.

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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QUESTION 2 Consider the heat distribution u (x, t) in a semi-infinite rod. Assume that the finite end is kept in contact with ice at 0°C and if initially the rod is submerged in hot water at 100°C. (a) Use the appropriate Fourier transform to show that (explain the choice of the Fourier transform) d û(a, t) + c²a²û (a, t) = 100ac² with Aû (a,0) = 0, dt where û is the appropriate Fourier transform of u and c is the specific heat of the rod.
Transcribed Image Text:QUESTION 2 Consider the heat distribution u (x, t) in a semi-infinite rod. Assume that the finite end is kept in contact with ice at 0°C and if initially the rod is submerged in hot water at 100°C. (a) Use the appropriate Fourier transform to show that (explain the choice of the Fourier transform) d û(a, t) + c²a²û (a, t) = 100ac² with Aû (a,0) = 0, dt where û is the appropriate Fourier transform of u and c is the specific heat of the rod.
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To consider the heat distribution ux,t in a semi-infinite rod.a To show that   ddtuα,t+c2α2u^α,t=100αc22π ,with Au^α,0=0where u^ is the appropriate fourier transform of u and c is the specific heat of the rod

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