Page 68 - IJOCTA-15-4

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.4, pp.610-624 (2025)
https://doi.org/10.36922/IJOCTA025130067
RESEARCH ARTICLE

Analysis and analytical solution of incommensurate fuzzy fractional
nabla difference systems in neural networks

1*
2
Babak Shiri , Ehsan Dadkhah Khiabani , and Dumitru Baleanu 3,4
1 Key Laboratory of Numerical Simulation for Sichuan Provincial Universities, College of Mathematics and
Information Science, Neijiang Normal University, Neijiang, China
2 Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran
3 Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
4 Institute of Space Sciences-subsidiary of INFLPR, Magurele-Bucharest, Romania
shiribabak2018@gmail.com, ehsan.dadkhah@gmail.com, dumitru.baleanu@gmail.com

ARTICLE INFO ABSTRACT

Article History:
Received: March 30, 2025 Uncertain incommensurate fractional nabla difference systems (IFDSs) in re-
1st revised: May 31, 2025 current neural networks (RNNs) are analyzed using fuzzy number theory to
2nd revised: June 12, 2025 address input uncertainties. Fuzzy number theory and its operations are re-
Accepted: June 25, 2025 investigated, and the H-differenceable concept is introduced. The existence of
Published Online: July 18, 2025 a unique H-differenceable solution for incommensurate RNNs is proved. A re-
cursive algorithm is proposed to obtain fuzzy solutions. Illustrative examples
Keywords:
with 2-dimensional IFDSs are provided to validate the framework for integrat-
Fuzzy numbers
ing fractional calculus, fuzzy dynamics and incommensurate RNNs.
Fuzzy neural networks
Incommensurate systems
Nabla fractional difference
Recurrent neural networks
Uncertainty analysis
AMS Classification 2010:
26A33; 92B20; 39A05; 15B15
46S40; 90C70

1. Introduction system, Al-Taani et al. 10 investigated a discrete
memristive model with incommensurate orders,
Discrete equations offer notable advantages over 11
and Shatnaei et al. studied the stability of non-
continuous-based models in specific scenarios,
linear incommensurate fractional order difference
such as digital signal processing and neural net-
systems. Despite these advances, there remains a
works. For nonlinear discrete models, the recur-
critical gap in addressing uncertainty in such sys-
sive nature of solutions can, under appropriate
tems, particularly when fuzzy inputs are involved,
conditions, eliminate the need for a separate ex- a limitation that motivates our work.
istence analysis. In addition, it introduces re-
cursive methods for obtaining exact or numerical In this paper, we focus on a type of dense re-
solutions. 1–5 current neural network (RNN) described by dis-
crete incommensurate nabla fractional difference
Recent research has witnessed a growing inter-
equations.
est in incommensurate fractional differential sys-
tems. However, the body of research on discrete Let x i : N 0 → R, i = 1, . . . , ν, be the inputs
incommensurate fractional difference systems re- of a neural network. These inputs can represent
mains relatively undiscovered (see Refs. 6–8 and signals or image information. Let w ij ∈ R de-
9
references therein). Abbes et al. explored an in- note the weights and p i denote the bias terms.
commensurate fractional discrete macroeconomic We consider an incommensurate fractional nabla
*Corresponding Author
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