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Suppose that in a semi-plate z ≥ 0, 0≤ y ≤ there is a heat source depending on the variable y and that the steady temperature u(x, y) satisfies the following Poisson's equation. Uxx (x, y) + Uyy (x, y) + s(y)=0, x > 0, 0

Suppose that in a semi-plate z ≥ 0, 0≤ y ≤ there is a heat source depending on the variable y and that the steady temperature u(x, y) satisfies the following Poisson's equation. Uxx (x, y) + Uyy (x, y) + s(y)=0, x > 0, 0

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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Suppose that in a semi-plate x ≥ 0, 0≤ y ≤ there is a heat source depending on the variable y and that the steady temperature u(x, y) satisfies the following Poisson's equation Uxx (x, y) + Uyy (x, y) + s(y)=0, x > 0, 0 <y<π, (1) and the following boundary conditions u(x, π) = 0, u(x,0) = 0, u(0, y) = 0 (2). Poisson's equation (1) is a non-homogeneous equation. By assuming for the steady temperature a bounded solution of the form: u(x, y) = 1 Bn(x) sin ny and using the method of variation of parameters, solve this boundary value problem Equations (1) and (2). -5) Note that the Fourier sine series of s(y) = Also assume that the temperature u(x, y) is to be bounded when x tends to ∞o. 1 Sn sin ny; Sn = Sn = s(y) sin nydy
Transcribed Image Text:Suppose that in a semi-plate x ≥ 0, 0≤ y ≤ there is a heat source depending on the variable y and that the steady temperature u(x, y) satisfies the following Poisson's equation Uxx (x, y) + Uyy (x, y) + s(y)=0, x > 0, 0 <y<π, (1) and the following boundary conditions u(x, π) = 0, u(x,0) = 0, u(0, y) = 0 (2). Poisson's equation (1) is a non-homogeneous equation. By assuming for the steady temperature a bounded solution of the form: u(x, y) = 1 Bn(x) sin ny and using the method of variation of parameters, solve this boundary value problem Equations (1) and (2). -5) Note that the Fourier sine series of s(y) = Also assume that the temperature u(x, y) is to be bounded when x tends to ∞o. 1 Sn sin ny; Sn = Sn = s(y) sin nydy
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