Neural networking study of worms in a wireless sensor model in the sense of fractal fractional

Aziz Khan; Thabet Abdeljawad; Manar A. Alqudah; Aziz Khan; Thabet Abdeljawad; Manar A. Alqudah

doi:10.3934/math.20231348

AIMS Mathematics

2023, Volume 8, Issue 11: 26406-26424. doi: 10.3934/math.20231348

Previous Article Next Article

Research article Special Issues

Neural networking study of worms in a wireless sensor model in the sense of fractal fractional

1.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia
2.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
3.
Department of Mathematics, Kyung Hee University 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea
4.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa
5.
Department of Mathematical Sciences, Faculty of Science, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 09 July 2023 Revised: 22 August 2023 Accepted: 31 August 2023 Published: 15 September 2023
MSC : 26A33, 34A12, 34A08, 34K12

We are concerned with the analysis of the neural networks of worms in wireless sensor networks (WSN). The concerned process is considered in the form of a mathematical system in the context of fractal fractional differential operators. In addition, the Banach contraction technique is utilized to achieve the existence and unique outcomes of the given model. Further, the stability of the proposed model is analyzed through functional analysis and the Ulam-Hyers (UH) stability technique. In the last, a numerical scheme is established to check the dynamical behavior of the fractional fractal order WSN model.
- fractal-fractional operator,
- neural networking,
- Mittag-Leffler kernel,
- Ulam-Hyers stability,
- Banach contraction,
- numerical analysis
Citation: Aziz Khan, Thabet Abdeljawad, Manar A. Alqudah. Neural networking study of worms in a wireless sensor model in the sense of fractal fractional[J]. AIMS Mathematics, 2023, 8(11): 26406-26424. doi: 10.3934/math.20231348

Related Papers:

Abstract

We are concerned with the analysis of the neural networks of worms in wireless sensor networks (WSN). The concerned process is considered in the form of a mathematical system in the context of fractal fractional differential operators. In addition, the Banach contraction technique is utilized to achieve the existence and unique outcomes of the given model. Further, the stability of the proposed model is analyzed through functional analysis and the Ulam-Hyers (UH) stability technique. In the last, a numerical scheme is established to check the dynamical behavior of the fractional fractal order WSN model.

References

[1]	P. S. Tippett, The kinetics of computer virus replication: A theory and preliminary survey, in Safe Computing: Proceedings of the Fourth Annual Computer Virus and Security Conference, 1991, 14–15.
[2]	J. R. Piqueira, F. B. Cesar, Dynamical models for computer viruses propagation, Math. Probl. Eng., 2008 (2008), 1–11. https://doi.org/10.1155/2008/940526 doi: 10.1155/2008/940526
[3]	J. O. Kephart, S. R. White, Directed-graph epidemiological models of computer viruses, Comput. Micro Macro View., 1992 (1992), 71–102. https://doi.org/10.1142/9789812812438-0004 doi: 10.1142/9789812812438-0004
[4]	F. Cohen, A cost analysis of typical computer viruses and defenses, Comput. Secur., 10 (1991), 239–250. https://doi.org/10.1016/0167-4048(91)90040-K doi: 10.1016/0167-4048(91)90040-K
[5]	M. E. J. Newman, S. Forrest, J. Balthrop, Email networks and the spread of computer viruses, Phys. Rev. E., 66 (2002), 1–4. https://doi.org/10.1103/PhysRevE.66.035101 doi: 10.1103/PhysRevE.66.035101
[6]	B. K. Mishra, D. Saini, Mathematical models on computer viruses, Appl. Math. Comput., 187 (2007), 929–936. https://doi.org/10.1016/j.amc.2006.09.062 doi: 10.1016/j.amc.2006.09.062
[7]	D. Bernoulli, D. Chapelle, Essai dune nouvelle analyse de la mortalite causee par la petite verole, et des avantages de linoculation pour la prevenir, Histoire de lAcad, Roy. Sci.(Paris) avec Mem., 1760 (1760), 1–45. https://inria.hal.science/hal-04100467
[8]	J. O. Kephart, S. R. White, D. M. Chess, Computers and epidemiology, IEEE Spectrum., 30 (1993), 20–26. https://doi.org/10.1109/6.275061 doi: 10.1109/6.275061
[9]	L. Billings, W. M. Spears, I. B. Schwartz, A unified prediction of computer virus spread in connected networks, Phys. Lett. A., 297 (2002), 261–266. https://doi.org/10.1016/S0375-9601(02)00152-4 doi: 10.1016/S0375-9601(02)00152-4
[10]	L. Zhang, S. Ahmad, A. Ullah, A. Akgãœl, E. K. Akgãœl, Analysis Of Hidden Attractors Of Non-Equilibrium Fractal-Fractional Chaotic System With One Signum Function, FRACTALS, 30 (2022), 1–6. https://doi.org/10.1142/S0218348X22401399 doi: 10.1142/S0218348X22401399
[11]	S. Etemad, B. Tellab, A. Zeb, S. Ahmad, A. Zada, S. Rezapour, et al., A mathematical model of transmission cycle of CC-Hemorrhagic fever via fractal–fractional operators and numerical simulations, Results Phys., 40 (2022), 105800. https://doi.org/10.1016/j.rinp.2022.105800 doi: 10.1016/j.rinp.2022.105800
[12]	H. Khan, J. Alzabut, D. Baleanu, G. Alobaidi, M. U. Rehman, Existence of solutions and a numerical scheme for a generalized hybrid class of n-coupled modified ABC-fractional differential equations with an application, AIMS Math., 8 (2023), 6609–6625. https://doi.org/10.3934/math.2023334 doi: 10.3934/math.2023334
[13]	H. Khan, J. Alzabut, A. Shah, Z. Y. He, S. Etemad, S. Rezapour, et al., On fractal-fractional waterborne disease model: A study on theoretical and numerical aspects of solutions via simulations, Fractals., 2023. https://doi.org/10.1142/S0218348X23400558 doi: 10.1142/S0218348X23400558
[14]	H. Khan, J. Alzabut, H. Gulzar, Existence of solutions for hybrid modified ABC-fractional differential equations with p-Laplacian operator and an application to a waterborne disease model, Alex. Eng. J., 70 (2023), 665–672. https://doi.org/10.1016/j.aej.2023.02.045 doi: 10.1016/j.aej.2023.02.045
[15]	A. Atangana, J. F. Gómez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu, Numer. Methods Partial Differ, Equations., 34 (2018), 1502–1523. https://doi.org/10.1002/num.22195 doi: 10.1002/num.22195
[16]	A. Akgül, I. M. Sajid, U. Fatima, N. Ahmed, Z. Iqbal, A. Raza, et al., Optimal existence of fractional order computer virus epidemic model and numerical simulations, Math. Methods Appl. Sci., 44 (2021), 10673–10685. https://doi.org/10.1002/mma.7437 doi: 10.1002/mma.7437
[17]	H. I. Abdel-Gawad, D. Baleanu, A. H. Abdel-Gawad, Unification of the different fractional time derivatives: An application to the epidemic-antivirus dynamical system in computer networks, Chaos Soliton. Fract., 142 (2021), 110416. https://doi.org/10.1016/j.chaos.2020.110416 doi: 10.1016/j.chaos.2020.110416
[18]	J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504–515. https://doi.org/10.1016/j.amc.2017.08.048 doi: 10.1016/j.amc.2017.08.048
[19]	N. Özdemir, S. Uçar, B. B. Eroglu, Dynamical analysis of fractional order model for computer virus propagation with kill signals, Int. J. Nonlinear Sci. Numer. Simul., 21 (2020), 239–247. https://doi.org/10.1515/ijnsns-2019-0063 doi: 10.1515/ijnsns-2019-0063
[20]	B. K. Mishra, N. Keshri, Mathematical model on the transmission of worms in wireless sensor network, Appl. Math. Model., 37 (2013), 4103–4111. https://doi.org/10.1016/j.apm.2012.09.025 doi: 10.1016/j.apm.2012.09.025
[21]	D. Kumar, J. Singh, New aspects of fractional epidemiological model for computer viruses with Mittag-Leffler law, Math. Model. Health Soc. Appl. Sci., 2020 (2020), 283–301. https://doi.org/10.1007/978-981-15-2286-4-9 doi: 10.1007/978-981-15-2286-4-9
[22]	A. Atangana, Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Soliton. Fract., 102 (2017), 396–406. https://doi.org/10.1016/j.chaos.2017.04.027 doi: 10.1016/j.chaos.2017.04.027
[23]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Sci. Ltd., 2006. https://shop.elsevier.com/books/theory-and-applications-of-fractional-differential-equations/kilbas/978-0-444-51832-3
[24]	T. Abdeljawad, D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equations., 2016 (2016), 1–22. https://doi.org/10.1186/s13662-016-0949-5 doi: 10.1186/s13662-016-0949-5
[25]	A. Khan, J. F. Gomez-Aguilar, T. S. Khan, H. Khan, Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chaos Soliton. Fract., 122 (2019), 119–128. https://doi.org/10.1016/j.chaos.2019.03.022 doi: 10.1016/j.chaos.2019.03.022
[26]	D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
[27]	T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. https://www.ams.org/journals/proc/1978-072-02/S0002-9939-1978-0507327-1
[28]	D. H. Hyers, T. M. Rassias, Approximate homomorphisms, Aequat. Math., 44 (1992), 125–153. https://doi.org/10.1007/BF01830975 doi: 10.1007/BF01830975

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)