4. i) The temperature in a bar of length L is described by the heat equation ди Ət for 0 0. მ2 The ends of the bar are insulated, which are boundary conditions with no heat flux, thus ди əx = 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).
4. i) The temperature in a bar of length L is described by the heat equation ди Ət for 0 0. მ2 The ends of the bar are insulated, which are boundary conditions with no heat flux, thus ди əx = 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).
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 A, Band C correctly in 30 minutes and get the thumbs up please show neat and clean work.
By hand solution needed
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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