Lecture09

# Lecture09 - Auxiliary Conditions and Solution of Separate...

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Unformatted text preview: Auxiliary Conditions and Solution of Separate ODEs The angular functions Q must be continuous and 2 π-periodic, and by Proposition 1 , such solutions are found for ν = n ∈ N (this also justifies our choice of a non- negative second separation constant): Q 1 n ( ϕ ) = cos( nϕ ) , Q 2 n ( ϕ ) = sin( nϕ ) , n ∈ N . (330) For the radial equation ( 328 ), we use the transformation z := kr and write w ( z ) := H ( r ). Then the ODE ( 328 ) transforms to z 2 w ′′ + zw ′ + ( z 2 − ν 2 ) w = 0 , z > . (331) Equation ( 331 ) is Bessel’s differential equation with parameter ν > 0. For integer values of the parameter, ν = n ∈ N , a fundamental system of solu- tions is given by the Bessel functions of the first and second kind, { J n ,Y n } , n ∈ N . Only the Bessel functions of the first kind are bounded as z → 0. The boundary conditions ( 322 ) require that J n ( kR ) = 0, n ∈ N . We obtain infinitely many solutions which involve the values k mn > 0, m ∈ N , n ∈ N . They satisfy J n ( k mn R ) = 0 . (332) Notice that the values k mn need to be approximated numerically (or found in a table) in practice. Remark: If we choose the first separation constant to be positive, we find the modified Bessel functions of the first and second kind, { I n ,K n } , n ∈ N . 58 Only the functions I n are bounded as z → 0. But they do not have any positive zeros, so they cannot satisfy the boundary condition at r = R . The case of a zero separation constant is left as an exercise (Problem Set 5). We obtain the following solutions for the radial equation ( 328 ): H mn ( r ) = J n ( k mn r ) , m ∈ N , n ∈ N . (333) Therefore, H mn is the n-th order Bessel function of the first kind, scaled such that its m-th zero is located at r = R , m ∈ N , n ∈ N . This implies that H mn has m − 1 zeros inside the domain r < R . We illustrate this for n = 1 , 2, m = 1 ,..., 4, R = 10, in the following figures: 59 The functions...
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Lecture09 - Auxiliary Conditions and Solution of Separate...

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