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

TL;DR

The paper tackles solving a Caputo fractional differential equation with a modified Bessel function source by first deriving the exact α=1/2 solution as a product of two modified Bessel functions. To overcome computational cost in evaluating Bessel functions, it develops a six-parameter extended MPQA that blends small- and large-argument information via a rational–hyperbolic ansatz, with two parameters (q,λ) optimized under positivity constraints. The approach achieves sub-percent relative errors (down to ~0.18% for the full solution) across ν∈[0,1], using only six fitting parameters, and it avoids numerical defects common in Padé-type approximations. The method provides a fast, accurate, and generalizable analytic tool for fractional nonhomogeneous problems and can be extended to other special functions arising in similar contexts.

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.

Figures (4)

• Figure 1: Dependence of the parameter $q$ on $\lambda$ and $\nu$. The physically admissible region corresponds to $q > 0$, ensuring regularity of the approximation function.
• Figure 2: Relative global error (\ref{['globalerror']}) as a function of the optimization parameter $\lambda$ and $\nu$ for the fitting function
• Figure 3: (a) shows the global minimum error as a function of $\nu$, where for each fixed $\nu$, the optimal $\lambda$ parameter is modeled by $\nu = 24.5(0.265 - \lambda)$. In Fig. (b), we can observe that using this linear approximation, the values of the $q$ parameter remain positive for all $\nu$.
• Figure 4: The relative punctual error (\ref{['punctualerror']}) as a function of the independent variable $x$ for the general solution (\ref{['finalsolutionexpample']}).