On construction of transmutation operators for perturbed Bessel equations

TL;DR

This paper develops a new series-based method for constructing transmutation operators for perturbed Bessel equations, providing explicit kernel representations, establishing new properties, and demonstrating numerical applications with uniform error bounds.

Contribution

It introduces a novel functional series representation for the transmutation kernel and a new uniform error bound for solutions of perturbed Bessel equations.

Findings

Explicit series representation of the transmutation kernel

New properties of the transmutation kernel established

Numerical application demonstrating uniform error bounds

Abstract

A representation for the kernel of the transmutation operator relating the perturbed Bessel equation with the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of the regular solution of the perturbed Bessel equation is given presenting a remarkable feature of uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of application to the solution of Dirichlet spectral problems is presented.