Publication Date:
2015-08-05
Description:
The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval R ≤ r ≤ γ R with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function Φ n , ν ( r ) = Y ν ′ ( λ n , ν ) J ν ( λ n , ν r / R ) - J ν ′ ( λ n , ν ) Y ν ( λ n , ν r / R ) or linear combinations of the spherical Bessel functions ψ m , ν ( r ) = y ν ′ ( λ m , ν ) j ν ( λ m , ν r / R ) - j ν ′ ( λ m , ν ) y ν ( λ m , ν r / R ) . The orthogonality relations with analytical expressions for the normalization constant are given. Explicit expressions for the Lommel integrals in terms of Lommel functions are derived. The cross product zeros Y ν ′ ( λ n , ν ) J ν ′ ( γ λ n , ν ) - J ν ′ ( λ n , ν ) Y ν ′ ( γ λ n , ν ) = 0 and y ν ′ ( λ m , ν ) j ν ′ ( γ λ m , ν ) - j ν ′ ( λ m , ν ) y ν ′ ( γ λ m , ν ) = 0 are considered in the complex plane for real as well as complex values of the index ν and approximations for the exceptional zero λ 1 , ν are obtained. A numerical scheme based on the discretization of the two-dimensional and three-dimensional Laplace operator with Neumann boundary conditions is presented. Explicit representations of the radial part of the Laplace operator in form of a tridiagonal matrix allow the simple computation of the cross product zeros.
Electronic ISSN:
2193-1801
Topics:
Natural Sciences in General
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