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Chem Differential Eq HW Solutions Fall 2011 66

Chem Differential Eq HW Solutions Fall 2011 66 - 66 Chapter...

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66 Chapter 4 Partial Differential Equations in Polar and Cylindrical Coordinates which shows that (1 - α 2 mn ) A 0 ,n = a 0 ,n is the n th Bessel coefficient of the Bessel series expansion of order 0 of the function 1. This series is computed in Example 1, Section 4.8. We have 1 = n =1 2 α 0 ,n J 1 ( α 0 ,n ) J 0 ( α 0 ,n r ) 0 <r< 1 . Hence (1 - α 2 mn ) A 0 ,n = 2 α 0 ,n J 1 ( α 0 ,n ) A 0 ,n = 2 (1 - α 2 mn ) α 0 ,n J 1 ( α 0 ,n ) ; and so u ( r,θ ) = n =1 2 (1 - α 2 mn ) α 0 ,n J 1 ( α 0 ,n ) J 0 ( α 0 ,n r ) . 9. Let h ( r ) = r if 0 <r< 1 / 2 , 0 if 1 / 2 <r< 1 . Then the equation becomes 2 u = f ( r,θ ), where f ( r,θ ) = h ( r ) sin θ . We proceed as in the previous exercise and try u ( r,θ ) = m =0 n =1 J m ( λ mn r )( A mn cos + B mn sin ) = m =0 n =1 φ mn ( r,θ ) , where φ mn ( r,θ ) = J m ( λ mn r )( A mn cos + B mn sin ). We plug this solution into the equation, use the fact that 2 ( φ mn ) = - λ 2 mn φ mn = - α 2 mn φ mn , and get 2 m =0 n =1 φ mn ( r,θ ) = h ( r ) sin θ m =0 n =1 2 ( φ mn ( r,θ )) = h ( r ) sin θ m =0 n =1 - α 2 mn φ mn ( r,θ ) = h ( r ) sin θ. We recognize this expansion as the expansion of the function
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