Example 2: Find the steady-state temperature u(r,0) in a semicircular plate. The boundary value problem is a²u¸ 1 ðu +-+ ər² ' r ər ' r2 a02 1 a²u = 0, 0<0 < T, 0
Example 2: Find the steady-state temperature u(r,0) in a semicircular plate. The boundary value problem is a²u¸ 1 ðu +-+ ər² ' r ər ' r2 a02 1 a²u = 0, 0<0 < T, 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:
\[
\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0, \quad 0 < \theta < \pi, \quad 0 < r < c
\]
Boundary conditions:
\[
u(c, \theta) = u_0, \quad 0 < \theta < \pi
\]
\[
u(r, 0) = 0, \quad u(r, \pi) = 0, \quad 0 < r < c
\]
This problem involves solving a partial differential equation to find the temperature distribution in a semicircular region, considering specific boundary conditions at the edge of the plate and along its diameter.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b020904-503e-46f0-a83a-8c74ba785f01%2F2e3b043f-8f4e-4860-aeca-268dd78ed4d4%2F50e9g6p_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:
\[
\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0, \quad 0 < \theta < \pi, \quad 0 < r < c
\]
Boundary conditions:
\[
u(c, \theta) = u_0, \quad 0 < \theta < \pi
\]
\[
u(r, 0) = 0, \quad u(r, \pi) = 0, \quad 0 < r < c
\]
This problem involves solving a partial differential equation to find the temperature distribution in a semicircular region, considering specific boundary conditions at the edge of the plate and along its diameter.
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