New formulation for discrete dynamical type inequalities via $ h $-discrete fractional operator pertaining to nonsingular kernel

Maysaa Al Qurashi; Saima Rashid; Sobia Sultana; Hijaz Ahmad; Khaled A. Gepreel; Maysaa Al Qurashi; Saima Rashid; Sobia Sultana; Hijaz Ahmad; Khaled A. Gepreel

doi:10.3934/mbe.2021093

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 2: 1794-1812. doi: 10.3934/mbe.2021093

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New formulation for discrete dynamical type inequalities via $ h $-discrete fractional operator pertaining to nonsingular kernel

1.
Department of Mathematics, King Saud University, P.O.Box 22452, Riyadh 11495, Saudi Arabia
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics and Statistics, Imam Muhammad Ibn Saud Islamic University, Riyadh, Saudi Arabia
4.
Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan
5.
Department of Mathematics, Faculty of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 18 December 2020 Accepted: 26 January 2021 Published: 20 February 2021

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 Grüss 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.
- discrete fractional calculus,
- Atangana-Baleanu fractional differences and sums,
- discrete Mittag-Leffler function,
- Grüss type inequality,
- Young inequality
Citation: Maysaa Al Qurashi, Saima Rashid, Sobia Sultana, Hijaz Ahmad, Khaled A. Gepreel. New formulation for discrete dynamical type inequalities via $ h $-discrete fractional operator pertaining to nonsingular kernel[J]. Mathematical Biosciences and Engineering, 2021, 18(2): 1794-1812. doi: 10.3934/mbe.2021093

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Abstract

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 Grüss 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.

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