4. Suppose that in the plate described in Example 2, Sec. 43, there is a heat source de- pending on the variable y and that the entire boundary is kept at temperature zero. According to Sec. 23, the steady temperatures u(x, y) in the plate must now satisfy Poisson's equation Uxx (x, y) + Uyy(x, y) +q(y) = 0 (a) By assuming a (bounded) solution of the form u(x, y) = u(x, y) n=1 of this temperature problem and using the method of variation of parameters (Sec. 42), show formally that = 8 B₁(x) = B₂ (x) sin ny (n = 1, 2,...), where qn are the coefficients in the Fourier sine series for q(y) on the interval 0 0,0
4. Suppose that in the plate described in Example 2, Sec. 43, there is a heat source de- pending on the variable y and that the entire boundary is kept at temperature zero. According to Sec. 23, the steady temperatures u(x, y) in the plate must now satisfy Poisson's equation Uxx (x, y) + Uyy(x, y) +q(y) = 0 (a) By assuming a (bounded) solution of the form u(x, y) = u(x, y) n=1 of this temperature problem and using the method of variation of parameters (Sec. 42), show formally that = 8 B₁(x) = B₂ (x) sin ny (n = 1, 2,...), where qn are the coefficients in the Fourier sine series for q(y) on the interval 0 0,0
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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