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For the disk (use radius equal 1), explain that the guess u = T(t)R(r)0(0) is more appropriate,

For the disk (use radius equal 1), explain that the guess u = T(t)R(r)0(0) is more appropriate,

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
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Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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5) For the disk (use radius equal 1), explain that the guess u = and remind the reader that the boundary condition has two pieces: R(1) = 0, and O(0) must be 27 periodic. T(t)R(r)O(0) is more appropriate, 6) Quote the formula for Urx + Uyy in polar coordinates; we did not prove it in class and you do not have to prove it either. Show that, when you try to solve Helmholtz equation in polar coordinates, with the guess V = R(r)0(0), you get two separate ODE's for R and O.
Transcribed Image Text:5) For the disk (use radius equal 1), explain that the guess u = and remind the reader that the boundary condition has two pieces: R(1) = 0, and O(0) must be 27 periodic. T(t)R(r)O(0) is more appropriate, 6) Quote the formula for Urx + Uyy in polar coordinates; we did not prove it in class and you do not have to prove it either. Show that, when you try to solve Helmholtz equation in polar coordinates, with the guess V = R(r)0(0), you get two separate ODE's for R and O.
Expert Solution
Step 1

Solution a)

For a disk-shaped domain, the problem we want to solve is

urr +ur/r+ uθθ/r2 = 0, u(a, θ) = h(θ).

By using the boundary condition, where the radius = 1 and θ =2π

Separating variables u = R(r)Θ(θ) gives R''Θ + r−1 R'Θ + r−2 RΘ'' = 0 or after multiplying by r2/(RΘ),

Θ''/Θ = −r2R'' − rR'/R = −λ.

 

Since u(r, θ) is 2π periodic, Θ and its derivatives should be also. The eigenvalue problem to be solved for Θ is

Θ'' + λΘ = 0, Θ(0) = Θ(2π), Θ'(0) = Θ'(2π)

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