We introduce fractional-order Bessel functions (FBFs) to obtain an approximate solution for various kinds of differential equations. Our main aim is to consider the new functions based on Bessel polynomials to the fractional calculus. To calculate derivatives and integrals, we use Caputo fractional derivatives and Riemann-Liouville fractional integral definitions. Then, operational matrices of fractional-order derivatives and integration for FBFs are derived. Also, we discuss an error estimate between the computed approximations and the exact solution and apply it in some examples. Applications are given to three model problems to demonstrate the effectiveness of the proposed method.

中文翻译：

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具有各种应用的分数阶贝塞尔函数

我们引入分数阶贝塞尔函数 (FBF) 来获得各种微分方程的近似解。我们的主要目标是考虑基于贝塞尔多项式的新函数到分数阶微积分。为了计算导数和积分，我们使用 Caputo 分数阶导数和 Riemann-Liouville 分数阶积分定义。然后，导出了 FBF 的分数阶导数和积分的运算矩阵。此外，我们讨论了计算近似值和精确解之间的误差估计，并将其应用于一些示例。给出了三个模型问题的应用，以证明所提出方法的有效性。