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
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+](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F26df40ab-87bd-48a7-b665-7a45f0ce31b9%2F6ae4b5e1-e521-49b6-9a9d-adadd6e61bed%2Fes6zume_processed.png&w=3840&q=75)
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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