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

TL;DR

This paper introduces an iterative integral operator method to generate polynomial approximations of Bessel functions of the first kind, which are useful for both computational purposes and modeling long-time decay in fractional diffusion problems.

Contribution

It develops a new iterative approach to approximate Bessel functions using polynomial seed functions, linking these approximations to solutions of fractional diffusion equations.

Findings

Iterative method produces increasingly accurate polynomial approximations.

Polynomials generated from the constant seed function effectively approximate Bessel functions.

The approach connects polynomial approximations to physical decay modes in fractional diffusion problems.

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.