5. Consider a metal rod with a temperature distribution (x, t) as a function of distance x along the rod and time t, which obeys the heat equation ae Ət = α a²A მე2 ს where a > 0 is a thermal diffusivity constant. The rod has a length l (so that x = [0, l]). Suppose that at the end x = 0 it is insulated so that no heat escapes, and at the end xl it is held at the temperature 0. = (a) Express the boundary conditions as equations. (b) Using the method of separation of variables, show that solutions of the differential equation, subject to the boundary conditions from part (a), can be written as (2k + 1)²²α 4l2 [ }] (x,t) = Σ bk cos (2k+1)πx COS exp 2l k=0 (c) Suppose that the metal rod has the initial condition (x, 0) = 0, independent of x. Find the coefficients {b} in this case. l Hint: COS [cos (7(2n + 1) z) cos COS π(2m+1) 2l l X бпт, dx = 28nm, n, mЄ N. 2
5. Consider a metal rod with a temperature distribution (x, t) as a function of distance x along the rod and time t, which obeys the heat equation ae Ət = α a²A მე2 ს where a > 0 is a thermal diffusivity constant. The rod has a length l (so that x = [0, l]). Suppose that at the end x = 0 it is insulated so that no heat escapes, and at the end xl it is held at the temperature 0. = (a) Express the boundary conditions as equations. (b) Using the method of separation of variables, show that solutions of the differential equation, subject to the boundary conditions from part (a), can be written as (2k + 1)²²α 4l2 [ }] (x,t) = Σ bk cos (2k+1)πx COS exp 2l k=0 (c) Suppose that the metal rod has the initial condition (x, 0) = 0, independent of x. Find the coefficients {b} in this case. l Hint: COS [cos (7(2n + 1) z) cos COS π(2m+1) 2l l X бпт, dx = 28nm, n, mЄ N. 2
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
Solute question c please
![5. Consider a metal rod with a temperature distribution (x, t) as a function of distance
x along the rod and time t, which obeys the heat equation
ae
Ət
= α
a²A
მე2 ს
where a > 0 is a thermal diffusivity constant. The rod has a length l (so that x = [0, l]).
Suppose that at the end x = 0 it is insulated so that no heat escapes, and at the end
xl it is held at the temperature 0.
=
(a) Express the boundary conditions as equations.
(b) Using the method of separation of variables, show that solutions of the differential
equation, subject to the boundary conditions from part (a), can be written as
(2k + 1)²²α
4l2
[ }]
(x,t) = Σ bk cos
(2k+1)πx
COS
exp
2l
k=0
(c) Suppose that the metal rod has the initial condition (x, 0) = 0, independent of
x. Find the coefficients {b} in this case.
l
Hint:
COS
[cos (7(2n + 1) z) cos
COS
π(2m+1)
2l
l
X
бпт,
dx = 28nm, n, mЄ N.
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69cce4ac-4bf6-4e6b-8636-bf160e045b58%2Fde7a2dc3-8792-4d7c-816e-c6a6a8da40c3%2Fnq2bbhp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. Consider a metal rod with a temperature distribution (x, t) as a function of distance
x along the rod and time t, which obeys the heat equation
ae
Ət
= α
a²A
მე2 ს
where a > 0 is a thermal diffusivity constant. The rod has a length l (so that x = [0, l]).
Suppose that at the end x = 0 it is insulated so that no heat escapes, and at the end
xl it is held at the temperature 0.
=
(a) Express the boundary conditions as equations.
(b) Using the method of separation of variables, show that solutions of the differential
equation, subject to the boundary conditions from part (a), can be written as
(2k + 1)²²α
4l2
[ }]
(x,t) = Σ bk cos
(2k+1)πx
COS
exp
2l
k=0
(c) Suppose that the metal rod has the initial condition (x, 0) = 0, independent of
x. Find the coefficients {b} in this case.
l
Hint:
COS
[cos (7(2n + 1) z) cos
COS
π(2m+1)
2l
l
X
бпт,
dx = 28nm, n, mЄ N.
2
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