1. Consider a semi-infinite two-dimensional metal plate, as represented by the shaded region in Figure 1 below, and let u(x, y) be its steady-state temperature. y 0 X Figure 1 A semi-infinite two-dimensional metal plate. ди = 0 along ду Suppose that the boundaries along y = 0 and y = 4 are insulated, so that we have them, and the temperature at x = 0 is held at f(y)°C, where f(y) = 100 -16). 4 x →∞0 Meanwhile, the temperature as x → ∞ is finite, that is, lim u(x, y) < ∞. Using the method of separation of variables, solve for u(x, y). [Hint 1: Think carefully about the range of the separation constant that would give non-zero mode solution. If you're not sure, analyse all possible cases: negative, zero and positive separation constant. Hint 2: The boundary condition at infinity can be used to discard some part of the solution that diverges as x → ∞. For example, if we have A(Inx)² + B, then the condition tells us that A must be zero, but it says nothing about B.] 4
1. Consider a semi-infinite two-dimensional metal plate, as represented by the shaded region in Figure 1 below, and let u(x, y) be its steady-state temperature. y 0 X Figure 1 A semi-infinite two-dimensional metal plate. ди = 0 along ду Suppose that the boundaries along y = 0 and y = 4 are insulated, so that we have them, and the temperature at x = 0 is held at f(y)°C, where f(y) = 100 -16). 4 x →∞0 Meanwhile, the temperature as x → ∞ is finite, that is, lim u(x, y) < ∞. Using the method of separation of variables, solve for u(x, y). [Hint 1: Think carefully about the range of the separation constant that would give non-zero mode solution. If you're not sure, analyse all possible cases: negative, zero and positive separation constant. Hint 2: The boundary condition at infinity can be used to discard some part of the solution that diverges as x → ∞. For example, if we have A(Inx)² + B, then the condition tells us that A must be zero, but it says nothing about B.] 4
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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![1. Consider a semi-infinite two-dimensional metal plate, as represented by the shaded region in
Figure 1 below, and let u(x, y) be its steady-state temperature.
y
0
X
Figure 1 A semi-infinite two-dimensional metal plate.
ди
= 0 along
ду
Suppose that the boundaries along y = 0 and y = 4 are insulated, so that we have
them, and the temperature at x = 0 is held at f(y)°C, where
f(y) = 100
-16).
4
x →∞0
Meanwhile, the temperature as x → ∞ is finite, that is, lim u(x, y) < ∞. Using the method of
separation of variables, solve for u(x, y).
[Hint 1: Think carefully about the range of the separation constant that would give non-zero mode
solution. If you're not sure, analyse all possible cases: negative, zero and positive separation
constant.
Hint 2: The boundary condition at infinity can be used to discard some part of the solution that
diverges as x → ∞. For example, if we have A(Inx)² + B, then the condition tells us that A must
be zero, but it says nothing about B.]
4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f858ee4-5878-4548-a5bc-a937719c2c74%2F415d3a05-7f3c-444e-abd5-a635fe699f81%2Fwx7fz1e_processed.png&w=3840&q=75)
Transcribed Image Text:1. Consider a semi-infinite two-dimensional metal plate, as represented by the shaded region in
Figure 1 below, and let u(x, y) be its steady-state temperature.
y
0
X
Figure 1 A semi-infinite two-dimensional metal plate.
ди
= 0 along
ду
Suppose that the boundaries along y = 0 and y = 4 are insulated, so that we have
them, and the temperature at x = 0 is held at f(y)°C, where
f(y) = 100
-16).
4
x →∞0
Meanwhile, the temperature as x → ∞ is finite, that is, lim u(x, y) < ∞. Using the method of
separation of variables, solve for u(x, y).
[Hint 1: Think carefully about the range of the separation constant that would give non-zero mode
solution. If you're not sure, analyse all possible cases: negative, zero and positive separation
constant.
Hint 2: The boundary condition at infinity can be used to discard some part of the solution that
diverges as x → ∞. For example, if we have A(Inx)² + B, then the condition tells us that A must
be zero, but it says nothing about B.]
4
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