On a novel iterative method to compute polynomial approximations to Bessel functions of the first kind and its connection to the solution of fractional diffusion/diffusion-wave problems

Abstract

Abstract. We present an iterative method to obtain approximations to Bessel functions of the first kind J p (x) (p > −1) via the repeated application of an integral operator to an initial seed function f 0 (x). The class of seed functions f 0 (x) leading to sets of increasingly accurate approximations f n (x) is considerably large and includes any polynomial. When the operator is applied once to a polynomial of degree s, it yields a polynomial of degree s + 2, and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree. We focus on the set of polynomial approximations generated from the seed function f 0 (x) = 1. This set of polynomials is not only useful for the computation of J p (x), but also from a physical point of view, as it describes the long-time decay modes of certain fractional diffusion and diffusion-wave problems.