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Unformatted text preview: We consider Dirichlet boundary value problems which are posed either inside or outside of a sphere with radius R > 0: our domain is either := { x R 3  x  < R } ( squiggleright interior problem), or (361) + := { x R 3  x  > R } ( squiggleright exterior problem). (362) Notice that is a bounded domain, whereas + is not. In both cases, the boundary value problem has the form u = 0 in , (363) u ( R,, ) = f ( , ) , [0 , 2 ) , [0 , ] , (364) lim r u ( r,, ) = 0 , [0 , 2 ) , [0 , ] . (365) Separation of Variables We write the unknown function in the form u ( r,, ) = F ( r ) G ( , ). We separate the radial variable from the angular variables by r 2 F F + 2 r F F = 1 G S 2 G = k R , (366) and the separate differential equations are given by r 2 F + 2 rF  kF = 0 , r < R or r > R, (367) S 2 G + kG = 0 , [0 , 2 ) , [0 , ] . (368) We solve for G first. For that purpose, we separate again, G ( , ) = H ( ) Q ( ), and we obtain H H = sin Q (sin Q ) + k sin 2 = 2 R . (369) The separate ODEs are now given by H + 2 H = 0 , [0 , 2 ) , (370) sin 2 Q + sin cos Q + ( k sin 2  2 ) Q = 0 , [0 , ] . (371) 64 Auxiliary Conditions and Solution of Separate ODEs 2 periodic solutions of ( 370 ) are found for...
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 Spring '08
 Staff
 Math

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