Chem Differential Eq HW Solutions Fall 2011 57

Chem Differential Eq HW Solutions Fall 2011 57 - ( u 1 ) rr...

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Section 4.3 Vibrations of a Circular Membrane: General Case 57 u = r 2 l , dv = r k +1 J k +1 ( r ) dr , and hence du =2 lr 2 l - 1 dr and v = r k +1 J k +1 ( r ): ± r k +1+2 l J k ( r ) dr = ± r 2 l [ r k +1 J k ( r )] dr = r 2 l r k +1 J k +1 ( r ) - 2 l ± r 2 l - 1 r k +1 J k +1 ( r ) dr = r k +1+2 l J k +1 ( r ) - 2 l ± r k +2 l J k +1 ( r ) dr = r k +1+2 l J k +1 ( r ) - 2 l ± r ( k +1)+1+2( l - 1) J k +1 ( r ) dr and so, by the induction hypothesis, we get ± r k +1+2 l J k ( r ) dr = r k +1+2 l J k +1 ( r ) - 2 l l - 1 ² n =0 ( - 1) n 2 n ( l - 1)! ( l - 1 - n )! r k +2 l - n J k + n +2 ( r )+ C = r k +1+2 l J k +1 ( r ) + l - 1 ² n =0 ( - 1) n +1 2 n +1 l ! ( l - ( n + 1))! r k +1+2 l - ( n +1) J k +( n +1)+1 ( r )+ C = r k +1+2 l J k +1 ( r )+ l ² m =1 ( - 1) m 2 m l ! ( l - m )! r k +1+2 l - m J k + m +1 ( r )+ C = l ² m =0 ( - 1) m 2 m l ! ( l - m )! r k +1+2 l - m J k + m +1 ( r )+ C, which completes the proof by induction for all integers k 0 and all l 0. 13. The proper place for this problem is in the next section, since its solution invovles solving a Dirichlet problem on the unit disk. The initial steps are similar to the solution of the heat problem on a rectangle with nonzero boundary data (Exercise 11, Section 3.8). In order to solve the problem, we consider the following two subproblems: Subproblem #1 (Dirichlet problem)
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Unformatted text preview: ( u 1 ) rr + 1 r ( u 1 ) r + 1 r 2 ( u 1 ) = 0 , < r < 1 , < 2 , u 1 (1 , ) = sin 3 , < 2 . Subproblem #2 (to be solved after Fnding u 1 ( r, ) from Subproblem #1) ( u 2 ) t = ( u 2 ) rr + 1 r ( u 2 ) r + 1 r 2 ( u 2 ) , < r < 1 , < 2 , t > , u 2 (1 , , t ) = 0 , < 2 , t > , u 2 ( r, , 0) =-u 1 ( r, ) , < r < 1 , < 2 . You can check, using linearity (or superposition), that u ( r, , t ) = u 1 ( r, ) + u 2 ( r, , t ) is a solution of the given problem. The solution of subproblem #1 follows immediately from the method of Sec-tion 4.5. We have u 2 ( r, ) = r 3 sin 3 ....
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

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