Some integral identities involving products of general solutions of Bessel's equation of integral order

TL;DR

This paper derives new integral identities involving products of Bessel function solutions, aiding spectral decomposition and simplifying nonlinear dynamical equations in cylindrical geometries.

Contribution

It introduces novel identities for products of Bessel solutions, enabling algebraic expression of integral quantities and expansion of solutions in Fourier-Bessel series.

Findings

Derived identities for products of two and three Bessel solutions.

Facilitated algebraic computation of energy and helicity in plasma physics.

Enabled transformation of PDEs into time evolution equations.

Abstract

Spectral decomposition of dynamical equations using curl-eigenfunctions has been extensively used in fluid and plasma dynamics problems using their orthogonality and completeness properties for both linear and non-linear cases. Coefficients of such expansions are integrals over products of Bessel functions in problems involving cylindrical geometry. In this paper, certain identities involving products of two and three general solutions of Bessel's equation have been derived. Some of these identities have been useful in the study of Turner relaxation of annular magnetized plasma [S.K.H. Auluck, Phys. Plasmas, 16, 122504, 2009], where quadratic integral quantities such as helicity and total energy were expressed as algebraic functions of the arbitrary constants of the general solution of Bessel's equation, allowing their determination by a minimization procedure. Identities involving products of three solutions enable expanding a product of two solutions in a Fourier-Bessel series of single Bessel functions facilitating transformation of partial differential equations representing non-linear dynamics problems into time evolution equations by eliminating spatial dependences. These identities are required in an ongoing investigation of fluctuation-driven coherent effects in a nonlinear dynamical system.