A novel numerical method for stochastic conformable fractional differential systems

Aisha F. Fareed; Emad A. Mohamed; Mokhtar Aly; Mourad S. Semary; Aisha F. Fareed; Emad A. Mohamed; Mokhtar Aly; Mourad S. Semary

doi:10.3934/math.2025345

AIMS Mathematics

2025, Volume 10, Issue 3: 7509-7525. doi: 10.3934/math.2025345

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Research article

A novel numerical method for stochastic conformable fractional differential systems

1.
Department of Electrical Engineering, College of Engineering, Prince Sattam bin Abdulaziz University, Al Kharj 11942, Saudi Arabia
2.
Facultad de Ingeniería, Arquitectura y Diseño, Universidad San Sebastián, Bellavista 7, Santiago 8420524, Chile
3.
Basic Sciences Department, Faculty of Engineering, Badr University in Cairo, Cairo 11829, Egypt
4.
Basic Engineering Sciences Department, Benha Faculty of Engineering, Benha University, Benha 13512, Egypt

Received: 07 January 2025 Revised: 05 March 2025 Accepted: 07 March 2025 Published: 31 March 2025
MSC : 34A08, 65L05, 60H10

This study introduced the conformable fractional discrete Temimi–Ansari method (CFDTAM), a novel numerical framework designed to solve fractional stochastic nonlinear differential equations with enhanced efficiency and accuracy. By leveraging the conformable fractional derivative (CFD), the CFDTAM unifies classical and fractional-order systems while maintaining computational simplicity. The method's efficacy was demonstrated through applications to a stochastic population model and the Brusselator system, showcasing its ability to handle nonlinear dynamics with high precision. A comprehensive convergence analysis was also conducted to validate the reliability and stability of the proposed method. All computations were performed using Mathematica 12 software, ensuring accuracy and consistency in numerical simulations. CFDTAM sets a new benchmark in fractional stochastic modeling, paving the way for advancements in partial differential equations, delay systems, and hybrid models.
- stochastic differential equations,
- white noise,
- fractional-order systems,
- population model,
- conformable derivative
Citation: Aisha F. Fareed, Emad A. Mohamed, Mokhtar Aly, Mourad S. Semary. A novel numerical method for stochastic conformable fractional differential systems[J]. AIMS Mathematics, 2025, 10(3): 7509-7525. doi: 10.3934/math.2025345

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Abstract

This study introduced the conformable fractional discrete Temimi–Ansari method (CFDTAM), a novel numerical framework designed to solve fractional stochastic nonlinear differential equations with enhanced efficiency and accuracy. By leveraging the conformable fractional derivative (CFD), the CFDTAM unifies classical and fractional-order systems while maintaining computational simplicity. The method's efficacy was demonstrated through applications to a stochastic population model and the Brusselator system, showcasing its ability to handle nonlinear dynamics with high precision. A comprehensive convergence analysis was also conducted to validate the reliability and stability of the proposed method. All computations were performed using Mathematica 12 software, ensuring accuracy and consistency in numerical simulations. CFDTAM sets a new benchmark in fractional stochastic modeling, paving the way for advancements in partial differential equations, delay systems, and hybrid models.

References

[1]	J. A. Machado, V. Kiryakova, Recent history of the fractional calculus: Data and statistics, De Gruyter, 2019, 1–22. https://doi.org/10.1515/9783110571622-001
[2]	W. Malesza, M. Macias, D. Sierociuk, Analytical solution of fractional variable order differential equations, J. Comput. Appl. Math., 348 (2019), 214–236. https://doi.org/10.1016/j.cam.2018.08.035 doi: 10.1016/j.cam.2018.08.035
[3]	I. Petráš, J. Terpák, Fractional calculus as a simple tool for modeling and analysis of long memory process in industry, Mathematics, 7 (2019), 511. https://doi.org/10.3390/math7060511 doi: 10.3390/math7060511
[4]	A. F. Fareed, M. S. Semary, H. N. Hassan, An approximate solution of fractional order Riccati equations based on controlled Picard's method with Atangana-Baleanu fractional derivative, Alex. Eng. J., 61 (2022), 3673–3678. https://doi.org/10.1016/j.aej.2021.09.009 doi: 10.1016/j.aej.2021.09.009
[5]	B. M. Aboalnaga, L. A. Said, A. H. Madian, A. S. Elwakil, A. G. Radwan, Cole bio-impedance model variations in daucus carota sativus under heating and freezing conditions, IEEE Access, 7 (2019), 113254–113263. https://doi.org/10.1109/ACCESS.2019.2934322 doi: 10.1109/ACCESS.2019.2934322
[6]	A. F. Fareed, M. A. Elsisy, M. S. Semary, M. T. M. M. Elbarawy, Controlled Picard's transform technique for solving a type of time fractional Navier-Stokes equation rresulting from iincompressible fluid flow, Int. J. Appl. Comput. Math., 8 (2022), 184. https://doi.org/10.1007/s40819-022-01361-x doi: 10.1007/s40819-022-01361-x
[7]	M. R. Homaeinezhad, A. Shahhosseini, High-performance modeling and discrete-time sliding mode control of uncertain non-commensurate linear time invariant MIMO fractional order dynamic systems, Commun. Nonlinear Sci., 84 (2020), 105200. https://doi.org/10.1016/j.cnsns.2020.105200 doi: 10.1016/j.cnsns.2020.105200
[8]	N. A. Khalil, L. A. Said, A. G. Radwan, A. M. Soliman, Generalized two-port network-based fractional order filters, AEU-Int. J. Electron. C., 104 (2019), 128–146. https://doi.org/10.1016/j.aeue.2019.01.016 doi: 10.1016/j.aeue.2019.01.016
[9]	O. Elwy, L. A. Said, A. H. Madian, A. G. Radwan, All possible topologies of the fractional-order Wien oscillator family using different approximation techniques, Circ. Syst. Signal Pr., 38 (2019), 3931–3951. https://doi.org/10.1007/s00034-019-01057-6 doi: 10.1007/s00034-019-01057-6
[10]	A. Allagui, T. J. Freeborn, A. S. Elwakil, M. E. Fouda, B. J. Maundy, A. G. Radwan, et al., Review of fractional-order electrical characterization of supercapacitors, J. Power Sources, 400 (2018), 457–467. https://doi.org/10.1016/j.jpowsour.2018.08.047 doi: 10.1016/j.jpowsour.2018.08.047
[11]	A. M. Abdelaty, A. T. Azar, S. Vaidyanathan, A. Ouannas, A. G. Radwan, Applications of continuous-time fractional order chaotic systems, In: Mathematical Techniques of Fractional Order Systems, Amsterdam: Elsevier, 2018,409–449. https://doi.org/10.1016/B978-0-12-813592-1.00014-3
[12]	I. Petráš, Fractional-order nonlinear systems: Modeling, analysis, and simulation, Berlin: Springer, 2011,103–184. https://doi.org/10.1007/978-3-642-18101-6_5
[13]	S. Deshpande, S. Vaidyanathan, A. T. Azar, A. Ouannas, Applications of fractional-order systems: Chaos and control, Amsterdam: Elsevier, 2020. https://doi.org/10.1007/978-1-84996-335-0
[14]	K. B. Oldham, J. Spanier, The fractional calculus: Theory and applications of differentiation and integration to arbitrary order, New York: Academic Press, 1974. https://doi.org/10.1016/s0076-5392%2809%29x6012-1
[15]	I. Podlubny, Fractional differential equations, New York: Academic Press, 1999, 1–340.
[16]	E. C. D. Oliveira, J. A. T. Machado, A review of definitions for fractional derivatives and integrals, Math. Probl. Eng., 2014, 238459. http://dx.doi.org/10.1155/2014/238459
[17]	H. M. Ahmed, M. A. Ragusa, Nonlocal controllability of Sobolev-type conformable fractional stochastic evolution inclusions with Clarke subdifferential, B. Malays. Math. Sci. So., 45 (2022), 3239–3253. https://doi.org/10.1007/s40840-022-01377-y doi: 10.1007/s40840-022-01377-y
[18]	W. L. Duan, H. Fang, C. Zeng, Second-order algorithm for simulating stochastic differential equations with white noises, Physica A, 525 (2019), 491–497. https://doi.org/10.1016/j.physa.2019.03.112 doi: 10.1016/j.physa.2019.03.112
[19]	A. F. Fareed, M. S. Semary, Stochastic improved Simpson for solving nonlinear fractional-order systems using product integration rules, Nonlinear Eng., 14 (2025), 20240070. https://doi.org/10.1515/nleng-2024-0070 doi: 10.1515/nleng-2024-0070
[20]	T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
[21]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85. Available from: https://digitalcommons.aaru.edu.jo/pfda/vol1/iss2/1.
[22]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel, theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.48550/arXiv.1602.03408 doi: 10.48550/arXiv.1602.03408
[23]	M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91 (1971), 134–147. https://doi.org/10.1007/BF00879562 doi: 10.1007/BF00879562
[24]	W. Wyss, Fractional diffusion equation, J. Math. Phys., 27 (1986), 2782–2785. https://doi.org/10.1063/1.527251 doi: 10.1063/1.527251
[25]	R. Hermann, Fractional calculus: An introduction for physicists, New Jersey: World Scientific, 2014, 1–500. https://doi.org/10.1142/8934
[26]	R. Khalil, M. Al Horani, A. Yousef, M. A. Sababheh, new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[27]	A. F. Fareed, M. T. M. Elbarawy, M. S. Semary, Fractional discrete Temimi-Ansari method with singular and nonsingular operators: Applications to electrical circuits, Adv. Contin. Discret. M., 5 (2023), 1–17. https://doi.org/10.1186/s13662-022-03742-4 doi: 10.1186/s13662-022-03742-4
[28]	M. S. Semary, M. T. M. Elbarawy, A. F. Fareed, Discrete Temimi-Ansari method for solving a class of stochastic nonlinear differential equations, AIMS Math., 7 (2022), 5093–5105. https://doi.org/10.3934/math.2022283 doi: 10.3934/math.2022283
[29]	H. Temimi, A. R. Ansari, A computational iterative method for solving nonlinear ordinary differential equations, LMS J. Comput. Math., 18 (2015), 730–753. https://doi.org/10.1112/S1461157015000285 doi: 10.1112/S1461157015000285
[30]	M. A. Jawary, S. A. Hatif, Semi-analytical iterative method for solving differential-algebraic equations, Ain Shams Eng. J., 9 (2018), 2581–2586. https://doi.org/10.1016/j.asej.2017.07.004 doi: 10.1016/j.asej.2017.07.004
[31]	F. Ebrahimi, A. Hashemi, F. Ebrahimi, R. Mir, An iterative method for solving partial differential equations and solution of Korteweg-de Vries equations for showing the capability of the iterative method, World Appl. Program, 3 (2013), 320–327. https://doi.org/10.1016/j.amc.2011.03.084 doi: 10.1016/j.amc.2011.03.084
[32]	A. Arafa, A. E. Sayed, A. Hagag, A fractional Temimi-Ansari method (FTAM) with convergence analysis for solving physical equations, Math. Method. Appl. Sci., 44 (2021), 6612–6629. https://doi.org/10.1002/mma.7212 doi: 10.1002/mma.7212
[33]	Z. Odibat, S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett., 21 (2008), 194–199. https://doi.org/10.1016/j.aml.2007.02.022 doi: 10.1016/j.aml.2007.02.022
[34]	A. Noor, M. Bazuhair, M. E. Beltagy, Analytical and computational analysis of fractional stochastic models using iterated itô integrals, Fractal Fract., 7 (2023), 575. https://doi.org/10.3390/fractalfract7080575 doi: 10.3390/fractalfract7080575
[35]	C. Kelley, Solving nonlinear equations with Newton's method, USA, Philadelphia: PA, 2003. https://doi.org/10.1137/1.9780898718898
[36]	A. Noor, A. Barnawi, R. Nour, A. Assiri, M. E. Beltagy, Analysis of the stochastic population model with random parameters, Entropy, 22 (2020), 562. https://doi.org/10.3390/e22050562 doi: 10.3390/e22050562
[37]	J. Giet, P. Vallois, S. W. Mezieres, The logistic SDE, Theory Stoch. Pro., 20 (2015), 28–62. Available from: https://www.mathnet.ru/eng/thsp95.
[38]	K. Nouri, H. Ranjbar, D. Baleanu, L. Torkzadeh, Investigation on Ginzburg-Landau equation via a tested approach to Benchmark stochastic Davis-Skodje system, Alex. Eng. J., 60 (2021), 5521–5526. https://doi.org/10.1016/j.aej.2021.04.040 doi: 10.1016/j.aej.2021.04.040

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