Example 2: Find the steady-state temperature u(r,0) in a semicircular plate. The boundary value problem is a²u 1du 1 a²u + r ər Tr2aA2 = 0, 0<0
Example 2: Find the steady-state temperature u(r,0) in a semicircular plate. The boundary value problem is a²u 1du 1 a²u + r ər Tr2aA2 = 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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![**Example 2:** Find the steady-state temperature \( u(r, \theta) \) in a semicircular plate. The boundary value problem is
\[
\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0, \quad 0 < \theta < \pi, \quad 0 < r < c
\]
\[
u(c, \theta) = u_0, \quad 0 < \theta < \pi
\]
\[
u(r, 0) = 0, \quad u(r, \pi) = 0, \quad 0 < r < c
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2140facb-0981-40cd-bf23-81e224855fc4%2F25f16f00-01ee-46f0-8b50-988ef8ed2ffd%2F82sy786_processed.png&w=3840&q=75)
Transcribed Image Text:**Example 2:** Find the steady-state temperature \( u(r, \theta) \) in a semicircular plate. The boundary value problem is
\[
\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0, \quad 0 < \theta < \pi, \quad 0 < r < c
\]
\[
u(c, \theta) = u_0, \quad 0 < \theta < \pi
\]
\[
u(r, 0) = 0, \quad u(r, \pi) = 0, \quad 0 < r < c
\]
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