M257-316Notes_Lecture29

M257-316Notes_Lecture29 - Chapter 25 Lecture 29 More Wedge...

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Chapter 25 Lecture 29 More Wedge Problems Example 25.1 A wedge with Inhomogeneous BC u rr + 1 r u r + 1 r 2 u θθ =0 0 <r<a, 0 <θ<α (25.1) u ( r, 0) = u 0 u ( r, α )= u 1 u ( r, θ ) < as r 0 u ( a, θ f ( θ ) . (25.2) Let us look for the simplest function of θ only that satisFes the inhomoge- neous BC of the from: w ( θ )=( u 1 u 0 ) θ α + u 0 . Note that w θθ = 0 and that w (0) = u 0 and w ( α u 1 . Then let u ( r, θ w ( θ )+ v ( r, θ ). u rr + 1 r u r + 1 r 2 u θθ = v rr + 1 r v r + 1 r 2 v θθ v ( 0) = 0 v ( r, α )=0 v ( a, θ f ( θ ) w ( θ ) Essentially the problem solved in Example 24.2 (25.3) The solution is u ( r, θ u 1 u 0 ) θ α + u 0 + X n =1 c n r ( α ) sin µ nπθ α (25.4) 147
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Lecture 29 More Wedge Problems where c n = 2 α a ( α ) Z 0 ± f ( θ ) w ( θ ) ¤ sin µ nπθ α dθ. (25.5) Example 25.2 A wedge with insulating BC on θ = 0 and θ = α< 2 π . u rr + 1 r u r + 1 r 2 u θθ =0 u θ ( r, 0) = 0 u θ ( r, α )=0 u ( a, θ )= f ( θ ) . (25.6) Let u ( r, θ R ( r )Θ( θ ) r 2 µ R 0 + 1 r R 0 /R ( r Θ 0 / Θ= λ 2 (25.7) Θ equation i Θ 0 + λ 2 Θ=0 Θ 0 (0)=0=Θ 0 ( α ) ¾ Θ( θ A cos λθ + B sin( λθ ) Θ 0 (0) = λ =0or B =0; (25.8)
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This note was uploaded on 05/04/2011 for the course MATH 25 taught by Professor Lo during the Spring '11 term at BC.

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M257-316Notes_Lecture29 - Chapter 25 Lecture 29 More Wedge...

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