Solve the 2D Laplace's equation in polar coordinates for the equilibrium temperature u(r, 0) in a circular disk of radius r = 1 with the following boundary conditions: u(1,0) = f(0) (prescribed temperature). Hint: use separation of variables! V²u Laplace's equation in polar coordinates is given by: 18 ?u rər Ər = 7²-0 - u(1,0) = f(0) + 1 2²u r² 20² = 0.
Solve the 2D Laplace's equation in polar coordinates for the equilibrium temperature u(r, 0) in a circular disk of radius r = 1 with the following boundary conditions: u(1,0) = f(0) (prescribed temperature). Hint: use separation of variables! V²u Laplace's equation in polar coordinates is given by: 18 ?u rər Ər = 7²-0 - u(1,0) = f(0) + 1 2²u r² 20² = 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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![Solve the 2D Laplace’s equation in polar coordinates for the equilibrium temperature \( u(r, \theta) \) in a circular disk of radius \( r = 1 \) with the following boundary conditions: \( u(1, \theta) = f(\theta) \) (prescribed temperature). Hint: use separation of variables!
**Diagram:**
- A circle representing a circular disk.
- Inside the circle, it is labeled \(\nabla^2 u = 0\).
- On the boundary of the circle, it is labeled \(u(1, \theta) = f(\theta)\).
Laplace’s equation in polar coordinates is given by:
\[
\nabla^2 u = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0.
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Transcribed Image Text:Solve the 2D Laplace’s equation in polar coordinates for the equilibrium temperature \( u(r, \theta) \) in a circular disk of radius \( r = 1 \) with the following boundary conditions: \( u(1, \theta) = f(\theta) \) (prescribed temperature). Hint: use separation of variables!
**Diagram:**
- A circle representing a circular disk.
- Inside the circle, it is labeled \(\nabla^2 u = 0\).
- On the boundary of the circle, it is labeled \(u(1, \theta) = f(\theta)\).
Laplace’s equation in polar coordinates is given by:
\[
\nabla^2 u = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0.
\]
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