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Find the steady-state temperature u(r,θ) in a quarter semi-circular plate of radius r=5 subject to the heat equation in polar coordinates

Find the steady-state temperature u(r,θ) in a quarter semi-circular plate of radius r=5 subject to the heat equation in polar coordinates

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Find the steady-state temperature u(r,θ) in a quarter semi-circular plate of radius r=5 subject to the heat equation in polar coordinates

1 ди 1 ди + -- = 0 r dr dr? u(r, 0) = 0,0 <r< 5, u(r.) = 0,0 <r< 5, u(5, 0) = 5 sin(280), 0 < 0 < As in the circular plate model we can separate the PDE into two ODE's with r?R" + rR' – AR = 0 e" + 10 = 0 with O(0) = •()- We can solve the BVP in 0: 0(0) = (without a coefficient). Therefore vA is a multiple of an integer n, and 2 = Now that we know A, we can rewrite the differential equation in Ras (using the prime notation for derivatives): = 0. This is a Cauchy-Euler equation. Substituting the guessed solution r" into this equation we get the auxiliary equation = 0. The only bounded solution at r = 0 is R(r) = Therefore u(r, 0) = E, B. Note We don't need n = 0 since that term is zero. The coefficients B, are Fourier sine coefficients, therefore 2 B, = do if n = = 0 if n+
Transcribed Image Text:1 ди 1 ди + -- = 0 r dr dr? u(r, 0) = 0,0 <r< 5, u(r.) = 0,0 <r< 5, u(5, 0) = 5 sin(280), 0 < 0 < As in the circular plate model we can separate the PDE into two ODE's with r?R" + rR' – AR = 0 e" + 10 = 0 with O(0) = •()- We can solve the BVP in 0: 0(0) = (without a coefficient). Therefore vA is a multiple of an integer n, and 2 = Now that we know A, we can rewrite the differential equation in Ras (using the prime notation for derivatives): = 0. This is a Cauchy-Euler equation. Substituting the guessed solution r" into this equation we get the auxiliary equation = 0. The only bounded solution at r = 0 is R(r) = Therefore u(r, 0) = E, B. Note We don't need n = 0 since that term is zero. The coefficients B, are Fourier sine coefficients, therefore 2 B, = do if n = = 0 if n+
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