Qurashi Maysaa Al, Rashid Saima, Sultana Sobia, Ahmad Hijaz, Gepreel Khaled A
Department of Mathematics, King Saud University, P.O.Box 22452, Riyadh 11495, Saudi Arabia.
Department of Mathematics, Government College University, Faisalabad, Pakistan.
Math Biosci Eng. 2021 Feb 20;18(2):1794-1812. doi: 10.3934/mbe.2021093.
Discrete fractional calculus (DFC) use to analyse nonlocal behaviour of models has acquired great importance in recent years. The aim of this paper is to address the discrete fractional operator underlying discrete Atangana-Baleanu (AB)-fractional operator having $\hbar$-discrete generalized Mittag-Leffler kernels in the sense of Riemann type (ABR). In this strategy, we use the $\hbar$-discrete AB-fractional sums in order to obtain the Gr"{u}ss type and certain other related variants having discrete generalized $\hbar$-Mittag-Leffler function in the kernel. Meanwhile, several other variants found by means of Young, weighted-arithmetic-geometric mean techniques with a discretization are formulated in the time domain $\hbar\mathbb{Z}$. At first, the proposed technique is compared to discrete AB-fractional sums that uses classical approach to derive the numerous inequalities, showing how the parameters used in the proposed discrete $\hbar$-fractional sums can be estimated. Moreover, the numerical meaning of the suggested study is assessed by two examples. The obtained results show that the proposed technique can be used efficiently to estimate the response of the neural networks and dynamic loads.
近年来，用于分析模型非局部行为的离散分数阶微积分（DFC）变得极为重要。本文旨在探讨离散阿坦加纳 - 巴莱亚努（AB）分数阶算子背后的离散分数阶算子，该算子在黎曼型（ABR）意义下具有$\hbar$离散广义米塔格 - 莱夫勒核。在这一策略中，我们使用$\hbar$离散AB分数阶和，以得到核中具有离散广义$\hbar$米塔格 - 莱夫勒函数的格鲁斯型及其他一些相关变体。同时，通过杨不等式、加权算术 - 几何平均技术并结合离散化在时域$\hbar\mathbb{Z}$中给出了其他几种变体。首先，将所提出的技术与使用经典方法推导众多不等式的离散AB分数阶和进行比较，展示了如何估计所提出的离散$\hbar$分数阶和中使用的参数。此外，通过两个例子评估了所建议研究的数值意义。所得结果表明，所提出的技术可有效地用于估计神经网络和动态载荷的响应。
