1. Solve the following Laplace equation on a domain exterior to a circle: 1 r2U00 1 Au = Urr + −Ur + r ur (a,0) = 0, - 0 • > a, −π < 0 < π) lim u(r, 0) 3r cos 0 = 0. r→∞ (Hint: Note that this problem does not satisfy u(r, 0) < ∞ as r → ∞. To solve this problem, try to write u(r,0) = 3r cos 0 + v(r, 0) and solve the corresponding problem for v(r, 0).)
1. Solve the following Laplace equation on a domain exterior to a circle: 1 r2U00 1 Au = Urr + −Ur + r ur (a,0) = 0, - 0 • > a, −π < 0 < π) lim u(r, 0) 3r cos 0 = 0. r→∞ (Hint: Note that this problem does not satisfy u(r, 0) < ∞ as r → ∞. To solve this problem, try to write u(r,0) = 3r cos 0 + v(r, 0) and solve the corresponding problem for v(r, 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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![### Laplace Equation in a Circular Domain
**Problem Statement:**
Solve the following Laplace equation on a domain exterior to a circle:
\[
\Delta u = u_{rr} + \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta\theta} = 0 \quad (r > a, \ -\pi < \theta < \pi)
\]
**Boundary Conditions:**
1. \( u_r(a, \theta) = 0 \)
2. \(\lim_{r \to \infty} u(r, \theta) - 3r \cos{\theta} = 0 \)
**Hint:**
Note that this problem does not satisfy \( u(r, \theta) < \infty \) as \( r \to \infty \). To solve this problem, try to write \( u(r, \theta) = 3r \cos{\theta} + v(r, \theta) \) and solve the corresponding problem for \( v(r, \theta) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd3485286-e420-456a-8a41-5bc62e18ff5f%2F7a67bbcd-67cb-4234-9aa0-c035deb64085%2Ffrnn25_processed.png&w=3840&q=75)
Transcribed Image Text:### Laplace Equation in a Circular Domain
**Problem Statement:**
Solve the following Laplace equation on a domain exterior to a circle:
\[
\Delta u = u_{rr} + \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta\theta} = 0 \quad (r > a, \ -\pi < \theta < \pi)
\]
**Boundary Conditions:**
1. \( u_r(a, \theta) = 0 \)
2. \(\lim_{r \to \infty} u(r, \theta) - 3r \cos{\theta} = 0 \)
**Hint:**
Note that this problem does not satisfy \( u(r, \theta) < \infty \) as \( r \to \infty \). To solve this problem, try to write \( u(r, \theta) = 3r \cos{\theta} + v(r, \theta) \) and solve the corresponding problem for \( v(r, \theta) \).
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