Quasi-Rational Analytic Approximation for the Modified Bessel Function I1(x) with High Accuracy

Pablo Martin,Jorge Olivares,Eduardo Rojas,Adrián Sotomayor

doi:10.3390/sym13050741

Abstract

A new simple and accurate expression to approximate the modified Bessel function of the first kind I1(x) is presented in this work. This new approximation is obtained as an improvement of the multi-point quasi-rational approximation technique, MPQA. This method uses the power series of the Bessel function, its asymptotic expansion, and a process of optimization to fit the parameters of a fitting function. The fitting expression is formed by elementary functions combined with rational ones. In the present work, a sum of hyperbolic functions was selected as elementary functions to capture the first two terms of the asymptotic expansion of I1(x), which represents an important improvement with respect to previous research, where just the leading term of the asymptotic series was captured. The new approximation function presents a remarkable agreement with the analytical solution I1(x), decreasing the maximum relative error in more than one order of magnitude with respect to previous similar expressions. Concretely, the relative error was reduced from 10−2 to 4×10−4, opening the possibility of applying the new improved method to other Bessel functions. It is also remarkable that the new approximation is valid for all positive and negative values of the argument.

Highlights

The Bessel equation and its modified form play an important role for modeling common problems in science and engineering
These equations appear in different contexts spanning areas like electrodynamics [1,2], plasma-physics [3,4,5], heat transfer [6,7,8], chemical engineering [9], fluid mechanics [10,11], and others [12,13]
The criterion for determining λ is by comparing the maximum relative error of the fitting function for different λ values

Summary

Introduction

The Bessel equation and its modified form play an important role for modeling common problems in science and engineering. The numerical calculation for large and intermediate values of x requires a large number of terms of the series to get good accuracy, which has impulsed a range of different techniques for computing the modified Bessel functions properly in a reasonable time frame. These techniques pursue two kinds of goals. A first group focuses on formulating a simple approximation of the modified Bessel function with a low number of fitting parameters. They can be used in correlations originated from the Bessel equation or from more complex equations where the Bessel equation is involved

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Conclusion