Accurate analytic approximation for a fractional differential equation with a modified Bessel function term

TL;DR

This paper introduces a highly accurate analytical approximation for a fractional differential equation involving a modified Bessel function, utilizing an extended quasi-rational method to improve precision over existing models.

Contribution

The authors develop a novel approximation method that captures both series and asymptotic behaviors of Bessel functions, significantly enhancing accuracy for fractional differential equations.

Findings

Maximum relative error of approximately 0.18% with six parameters

The approximation accurately reproduces the asymptotic expansion

Method outperforms previous approximation techniques

Abstract

A new analytical approximation function is proposed to accurately fit the solution of a fractional differential equation of order one-half, whose nonhomogeneous term is defined by a modified Bessel function of the first kind. The exact analytical solution of this equation is expressed as the product of two modified Bessel functions. The approximation is constructed using an extended multipoint quasi-rational method, which simultaneously incorporates the series expansion and the asymptotic behavior of the Bessel function. A key modification is introduced in the structure of the fitting function, allowing it to reproduce two terms of the asymptotic expansion instead of only one, thereby improving accuracy for large arguments. Numerical analysis shows that for representative parameter values, the maximum relative error between the proposed fitting function and the exact solution of the fractional differential equation is approximately \(0.18\%\), demonstrating the high precision achieved with only six fitting parameters.