length, is given by the heat equation, ди 2²u əx² Ət (0 < x < 1, t > 0), here x measures distance along the bar, t is time, and u is temperature easured in ° C. Initially the bar has the temperature distribution (0 < x < 1). 2 u(x,0) = cos² 5πα 2 sin ( 5πα 2 he end of the bar at x = 0 is kept in contact with a block of ice at C, while the other end at x 1 is insulated. = O Write down the boundary conditions for the temperature distribution that model the situation described above. ) Use trigonometric identities to show that the initial condition can be expressed as u(x, 0) = P sin 5пх 7²) + 2 + Q sin (²¹ 15πα 2 and find the real numbers P and Q. Consider the eigenvalue problem X"(x) + XX(x) = 0, X(0)= X'(1) = 0. Find the eigenvalues and eigenfunctions. You may assume that all eigenvalues are positive. ) Use the method of separation of variables to find an expression for the temperature distribution in the bar for t > 0.
length, is given by the heat equation, ди 2²u əx² Ət (0 < x < 1, t > 0), here x measures distance along the bar, t is time, and u is temperature easured in ° C. Initially the bar has the temperature distribution (0 < x < 1). 2 u(x,0) = cos² 5πα 2 sin ( 5πα 2 he end of the bar at x = 0 is kept in contact with a block of ice at C, while the other end at x 1 is insulated. = O Write down the boundary conditions for the temperature distribution that model the situation described above. ) Use trigonometric identities to show that the initial condition can be expressed as u(x, 0) = P sin 5пх 7²) + 2 + Q sin (²¹ 15πα 2 and find the real numbers P and Q. Consider the eigenvalue problem X"(x) + XX(x) = 0, X(0)= X'(1) = 0. Find the eigenvalues and eigenfunctions. You may assume that all eigenvalues are positive. ) Use the method of separation of variables to find an expression for the temperature distribution in the bar for t > 0.
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
Just part (d)
Transcribed Image Text:The temperature distribution u(x, t) in a bar 1 m long, insulated along
its length, is given by the heat equation,
Ju J²u
Əx²
Ət
(0 < x < 1, t > 0),
where x measures distance along the bar, t is time, and u is temperature
measured in °C. Initially the bar has the temperature distribution
(0 < x < 1).
The end of the bar at x = 0 is kept in contact with a block of ice at
0° C, while the other end at x = 1 is insulated.
u(x, 0)
=
cos²
(577)
2
sin
(
5πx
2
(a) Write down the boundary conditions for the temperature
distribution that model the situation described above.
5πα
2
(b) Use trigonometric identities to show that the initial condition can
be expressed as
u(x, 0) = P sin
+ Q sin
(²
and find the real numbers P and Q.
(c) Consider the eigenvalue problem
15πα
X"(x) + XX(x) = 0,_X(0) = X′(1) = 0.
Find the eigenvalues and eigenfunctions. You may assume that all
eigenvalues are positive.
(d) Use the method of separation of variables to find an expression for
the temperature distribution in the bar for t > 0.
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