Page 80 - IJOCTA-15-4

B. Shiri et.al. / IJOCTA, Vol.15, No.4, pp.610-624 (2025)
Table 4. Exact output end local prediction Funding

{t} {t} This research is supported by the Neijiang Nor-
t y i x i (T) mal University school-level science and technol-
t = 341 0.8416 0.8489
ogy project (key project, No. XJ2024008301) and
−0.4955 −0.4751
Sharestan retrofit design co.
t = 271 0.7133 0.6651
−0.4711 −0.4489
Conflict of interest
t = 82 −0.0731 −0.0891
−0.0730 −0.0818 The authors declare that they have no conflict of
interest to disclose.

6. Conclusion
Author contributions
We proposed incommensurate fractional nabla
Conceptualization: Babak Shiri
discrete systems for a type of RNNs. These sys-
Formal analysis: All authors
tems become classical neural networks when all
Investigation: All authors
orders are zero, and classical RNNs when all or-
Methodology: Babak Shiri, Ehsan Dadkhah Khi-
ders are one. abani, Dumitru Baleanu
Using fuzzy theory, we analyzed how input Writing–original draft: Babak Shiri, Ehsan Dad-
data uncertainty affects output data. First, we re- khah Khiabani
visited the definition of fuzzy numbers and added Writing–review & editing: Dumitru Baleanu
a condition for the uniqueness of their determinis-
tic parts. We then used shape functions (instead
of membership functions) to handle operations on Availability of data
fuzzy numbers. We found that most discrete frac- Data used in this research are available on de-
tional nabla difference equations have a unique mand from the authors.
H-difference solution.
The GH-difference was introduced because AI tools statement
the H-difference isn’t defined for many uncertain
numbers. However, our results show that in the All authors confirm that no AI tools were used in
context of difference equations, we might not need the preparation of this manuscript.
to extend the difference definition for other uncer-
tain numbers. So, we introduced the concept of References
“H-differenceable.” Whether this holds in contin-
1. Shiri B, Guang Y, Baleanu D. Inverse problems
uous cases is unknown and needs future study.
for discrete Hermite nabla difference equation.
Based on our analysis, we developed a
Appl Math Sci Eng. 2025;33(1):2431000.
programmable algorithm to find fuzzy solu-
http://dx.doi.org/10.1080/27690911.2024.2431000
tions for the incommensurate RNNs. Examples 2. Beig Mohamadi R, Khastan A, Nieto JJ,
showed that nonlinearity deforms the shape func- Rodr´ıguez-L´opez R. Discrete fractional calculus
tions—an interesting finding. for fuzzy-number-valued functions and some re-
We trained a 2D incommensurate NN for local sults on initial value problems for fuzzy fractional
time series prediction. However, training the al- difference equations. Inf Sci. 2022;618:1–13.
gorithms for higher-dimensional incommensurate http://dx.doi.org/ 10.1016/j.ins.2022.10.062
NNs require further dedicated study. The struc- 3. Huang LL, Baleanu D, Mo ZW, Wu GC.
ture and derivatives of these NNs rely on recur- Fractional discrete-time diffusion equation with
uncertainty: applications of fuzzy discrete
sive calculations with previous datasets, making
fractional calculus. Phys A Stat Mech Appl.
higher recursion levels computationally complex.
2018;508:166–175.
This suggests that higher recursion may not be
http://dx.doi.org/ 10.1016/j.physa.2018.03.092
efficient for training. Additionally, our first ex-
4. Dassios IK. Stability and robustness of singular
ample showed that small input data inaccuracies systems of fractional nabla difference equations.
can cause results to deviate from expected values, Circuits Syst Signal Process. 2017;36:49–64.
implying higher recursion might not be effective http://dx.doi.org/ 10.1007/s00034-016-0291-x
either. 5. Dassios IK. A practical formula of solutions
for a family of linear nonautonomous fractional
Acknowledgments nabla difference equations. J Comput Appl Math.
2018;339:317–328.
None. http://dx.doi.org/ 10.1016/j.cam.2017.09.030
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