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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

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
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
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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