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i) The temperature in a bar of length L is described by the heat equation ди Ət Q202 əx² ди əx = for 0 0. The ends of the bar are insulated, which are boundary conditions with no heat flux, thus 0 at x = 0 and at x = L for t > 0. a) Using the method of separation of variables, let u(x, t) = X(x)T(t) and derive the following ODES X" = KX, T' = KQ²T. b) Consider all possibilities of the constant of separation and find all the non-trivial solutions X that satisfies the boundary conditions (

i) The temperature in a bar of length L is described by the heat equation ди Ət Q202 əx² ди əx = for 0 0. The ends of the bar are insulated, which are boundary conditions with no heat flux, thus 0 at x = 0 and at x = L for t > 0. a) Using the method of separation of variables, let u(x, t) = X(x)T(t) and derive the following ODES X" = KX, T' = KQ²T. b) Consider all possibilities of the constant of separation and find all the non-trivial solutions X that satisfies the boundary conditions (

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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Needed to be solved part I, A and B correctly in 30 minutes and get the thumbs up please show neat and clean work. By hand solution needed
Start a new page clearly marked Question 4 i) The temperature in a bar of length L is described by the heat equation ди Ət 4. Q20²u əx² ди əx e) Calculate = a The ends of the bar are insulated, which are boundary conditions with no heat flux, thus for 0<x< L and t > 0. 0 at x = 0 and at x = L for t> 0. a) Using the method of separation of variables, let u(x, t) = = X(x)T(t) and derive the following ODES X" = KX, T' = kQ²T. b) Consider all possibilities of the constant of separation and find all the non-trivial solutions X that satisfies the boundary conditions ( c) Find all possible solutions T, (t) for T(t). d) Hence find the solution u(x, t) if the initial temperature of the bar is 1, 0<x< 1/1, u(x,0) = q 0, ≤x≤L. steady state temperature of the bar.
Transcribed Image Text:Start a new page clearly marked Question 4 i) The temperature in a bar of length L is described by the heat equation ди Ət 4. Q20²u əx² ди əx e) Calculate = a The ends of the bar are insulated, which are boundary conditions with no heat flux, thus for 0<x< L and t > 0. 0 at x = 0 and at x = L for t> 0. a) Using the method of separation of variables, let u(x, t) = = X(x)T(t) and derive the following ODES X" = KX, T' = kQ²T. b) Consider all possibilities of the constant of separation and find all the non-trivial solutions X that satisfies the boundary conditions ( c) Find all possible solutions T, (t) for T(t). d) Hence find the solution u(x, t) if the initial temperature of the bar is 1, 0<x< 1/1, u(x,0) = q 0, ≤x≤L. steady state temperature of the bar.
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