A new analytical algorithm for uncertain fractional differential equations in the fuzzy conformable sense

Tareq Eriqat; Rania Saadeh; Ahmad El-Ajou; Ahmad Qazza; Moa'ath N. Oqielat; Ahmad Ghazal; Tareq Eriqat; Rania Saadeh; Ahmad El-Ajou; Ahmad Qazza; Moa'ath N. Oqielat; Ahmad Ghazal

doi:10.3934/math.2024472

AIMS Mathematics

2024, Volume 9, Issue 4: 9641-9681. doi: 10.3934/math.2024472

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A new analytical algorithm for uncertain fractional differential equations in the fuzzy conformable sense

1.
Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt 19117, Jordan
2.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan

Received: 10 January 2024 Revised: 20 February 2024 Accepted: 21 February 2024 Published: 11 March 2024
MSC : 34A08, 34K36, 40G10

This paper aims to explore and examine a fractional differential equation in the fuzzy conformable derivative sense. To achieve this goal, a novel analytical algorithm is formulated based on the Laplace-residual power series method to solve the fuzzy conformable fractional differential equations. The methodology being used to discover the fuzzy solutions depends on converting the desired equations into two fractional crisp systems expressed in $ \wp $-cut form. The main objective of our algorithm is to transform the systems into fuzzy conformable Laplace space. The transformation simplifies the system by reducing its order and turning it into an easy-to-solve algorithmic equation. The solutions of three important applications are provided in a fuzzy convergent conformable fractional series. Both the theoretical and numerical implications of the fuzzy conformable concept are explored about the consequential outcomes. The convergence analysis and theorems of the developed algorithm are also studied and analyzed in this regard. Additionally, this article showcases a selection of results through the use of both two-dimensional and three-dimensional graphs. Ultimately, the findings of this study underscore the efficacy, speed, and ease of the Laplace-residual power series algorithm in finding solutions for uncertain models that arise in various physical phenomena.
- conformable fractional derivative,
- fuzzy fractional power series,
- fuzzy conformable Laplace transform,
- exact solution
Citation: Tareq Eriqat, Rania Saadeh, Ahmad El-Ajou, Ahmad Qazza, Moa'ath N. Oqielat, Ahmad Ghazal. A new analytical algorithm for uncertain fractional differential equations in the fuzzy conformable sense[J]. AIMS Mathematics, 2024, 9(4): 9641-9681. doi: 10.3934/math.2024472

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Abstract

This paper aims to explore and examine a fractional differential equation in the fuzzy conformable derivative sense. To achieve this goal, a novel analytical algorithm is formulated based on the Laplace-residual power series method to solve the fuzzy conformable fractional differential equations. The methodology being used to discover the fuzzy solutions depends on converting the desired equations into two fractional crisp systems expressed in $ \wp $-cut form. The main objective of our algorithm is to transform the systems into fuzzy conformable Laplace space. The transformation simplifies the system by reducing its order and turning it into an easy-to-solve algorithmic equation. The solutions of three important applications are provided in a fuzzy convergent conformable fractional series. Both the theoretical and numerical implications of the fuzzy conformable concept are explored about the consequential outcomes. The convergence analysis and theorems of the developed algorithm are also studied and analyzed in this regard. Additionally, this article showcases a selection of results through the use of both two-dimensional and three-dimensional graphs. Ultimately, the findings of this study underscore the efficacy, speed, and ease of the Laplace-residual power series algorithm in finding solutions for uncertain models that arise in various physical phenomena.

References

[1]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353. http://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
[2]	A. Qazza, R. Saadeh, On the analytical solution of fractional SIR epidemic model, Appl. Comput. Intell. S., 2023 (2023), 6973734. https://doi.org/10.1155/2023/6973734 doi: 10.1155/2023/6973734
[3]	M. Sanchez-Roger, M. D. Oliver-Alfonso, C. Sanchis-Pedregosa, Fuzzy logic and its uses in finance: a systematic review exploring its potential to deal with banking crises, Mathematics, 7 (2019), 1091. http://doi.org/10.3390/math7111091 doi: 10.3390/math7111091
[4]	M. Guo, X. Xue, R. Li, Impulsive functional differential inclusions and fuzzy population models, Fuzzy Set. Syst., 138 (2003), 601–615. https://doi.org/10.1016/S0165-0114(02)00522-5 doi: 10.1016/S0165-0114(02)00522-5
[5]	O. Kaleva, Fuzzy differential equations, Fuzzy Set. Syst., 24 (1987), 301–317. https://doi.org/10.1016/0165-0114(87)90029-7 doi: 10.1016/0165-0114(87)90029-7
[6]	M. Al-Smadi, O. A. Arqub, D. Zeidan, Fuzzy fractional differential equations under the Mittag-Leffler kernel differential operator of the ABC approach: theorems and applications, Chaos Soliton. Fract., 146 (2021), 110891. https://doi.org/10.1016/j.chaos.2021.110891 doi: 10.1016/j.chaos.2021.110891
[7]	A. Kandel, Fuzzy mathematical techniques with applications, Boston: Addison-Wesley, 1986.
[8]	P. Narayana, Rao, K. Reddy, S. Sangam, C. S. E. Sreenidhi, Application of fuzzy logic in financial markets for decision making, International Journal of Advanced Research in Computer Science, 8 (2017), 382–386. https://doi.org/10.26483/IJARCS.V8I3.3020 doi: 10.26483/IJARCS.V8I3.3020
[9]	Y. Chalco-Cano, H. Roman-Flores, On new solutions of fuzzy differential equations, Chaos Soliton. Fract., 38 (2008), 112–119. https://doi.org/10.1016/j.chaos.2006.10.043 doi: 10.1016/j.chaos.2006.10.043
[10]	S. Hasan, M. Al-Smadi, A. El-Ajou, S. Momani, S. Hadid, Z. Al-Zhour, Numerical approach in the Hilbert space to solve a fuzzy Atangana-Baleanu fractional hybrid system, Chaos Soliton. Fract., 143 (2021), 110506. https://doi.org/10.1016/j.chaos.2020.110506 doi: 10.1016/j.chaos.2020.110506
[11]	D. Dubois, H. Prade, Towards fuzzy differential calculus part 1: Integration of fuzzy mappings, Fuzzy Set. Syst., 8 (1982), 1–17. https://doi.org/10.1016/0165-0114(82)90025-2 doi: 10.1016/0165-0114(82)90025-2
[12]	R. P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertain, Nonlinear Anal. Theor., 72 (2010), 2859–2862. https://doi.org/10.1016/j.na.2009.11.029 doi: 10.1016/j.na.2009.11.029
[13]	O. S. Fard, M. Salehi, A survey on fuzzy fractional variational problems, J. Comput. Appl. Math., 271 (2014), 71–82. https://doi.org/10.1016/j.cam.2014.03.019 doi: 10.1016/j.cam.2014.03.019
[14]	J. Soolaki, O. S. Fard, A. H. Borzabadi, Generalized Euler-Lagrange equations for fuzzy fractional variational calculus, Math. Commun., 21 (2016), 199–218.
[15]	J. Zhang, G. Wang, X. Zhi, C. Zhou, Generalized Euler-Lagrange equations for fuzzy fractional variational problems under gH-Atangana-Baleanu differentiability, J. Funct. Space., 2018 (2018), 2740678. http://doi.org/10.1155/2018/2740678 doi: 10.1155/2018/2740678
[16]	A. K. Das, T. K. Roy, Solving some system of linear fuzzy fractional differential equations by Adomian decomposition method, Intern. J. Fuzzy Mathematical Archive, 12 (2017), 83–92. http://doi.org/10.22457/ijfma.v12n2a5 doi: 10.22457/ijfma.v12n2a5
[17]	S. Salahshour, T. Allahviranloo, S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun. Nonlinear Sci., 17 (2012), 1372–1381. https://doi.org/10.1016/j.cnsns.2011.07.005 doi: 10.1016/j.cnsns.2011.07.005
[18]	B. Bede, S. G. Gal, Almost periodic fuzzy-number-valued functions, Fuzzy Set. Syst., 147 (2004), 385–403. https://doi.org/10.1016/j.fss.2003.08.004 doi: 10.1016/j.fss.2003.08.004
[19]	S. S. Behzadi, B. Vahdani, T. Allahviranloo, S. Abbasbandy, Application of fuzzy Picard method for solving fuzzy quadratic Riccati and fuzzy Painlevé I equations, Appl. Math. Model., 40 (2016), 8125–8137. https://doi.org/10.1016/j.apm.2016.05.003 doi: 10.1016/j.apm.2016.05.003
[20]	R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A 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
[21]	A. Qazza, A. Burqan, R. Saadeh, Application of ARA-Residual power series method in solving systems of fractional differential equations, Math. Probl. Eng., 2022 (2022), 6939045. https://doi.org/10.1155/2022/6939045 doi: 10.1155/2022/6939045
[22]	P. D. Spanos, A. D. Matteo, Y. Cheng, A. Pirrotta, J. Li, Galerkin scheme-based determination of survival probability of oscillators with fractional derivative elements, J. Appl. Mech., 83 (2016), 121003. https://doi.org/10.1115/1.4034460 doi: 10.1115/1.4034460
[23]	Y. Luo, P. D. Spanos, J. Chen, Stochastic response determination of multi-dimensional nonlinear systems endowed with fractional derivative elements by the GE-GDEE, Int. J. Nonlin. Mech., 147 (2022), 104247. https://doi.org/10.1016/j.ijnonlinmec.2022.104 doi: 10.1016/j.ijnonlinmec.2022.104
[24]	Y. Luo, M. Z. Lyu, J. B. Chen, P. D. Spanos, Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise, Theor. Appl. Mech. Lett., 13 (2023), 100436. https://doi.org/10.1016/j.taml.2023.100436 doi: 10.1016/j.taml.2023.100436
[25]	A. B. M. Alzahrani, M. A. Abdoon, M. Elbadri, M. Berir, D. E. Elgezouli, A comparative numerical study of the symmetry chaotic jerk system with a hyperbolic sine function via two different methods, Symmetry, 15 (2023), 1991. https://doi.org/10.3390/sym15111991 doi: 10.3390/sym15111991
[26]	R. Edwan, R. Saadeh, S. Hadid, M. Al-Smadi, M. Momani, Solving time-space-fractional Cauchy problem with constant coefficients by finite-difference method, In: Computational mathematics and applications, Singapore: Springer, 25–64. https://doi.org/10.1007/978-981-15-8498-5_2
[27]	A. M. Qazza, R. M. Hatamleh, N. A. Alodat, About the solution stability of volterra integral equation with random kernel, Far East Journal Of Mathematical Sciences, 100 (2016), 671–680, https://doi.org/10.17654/ms100050671 doi: 10.17654/ms100050671
[28]	O. A. Arqub, M. Al-Smadi, Fuzzy conformable fractional differential equations: novel extended approach and new numerical solutions, Soft Comput., 24 (2020), 12501–12522. https://doi.org/10.1007/s00500-020-04687-0 doi: 10.1007/s00500-020-04687-0
[29]	A. Harir, S. Melliani, L. S. Chadli, Analytic solution method for fractional fuzzy conformable Laplace transforms, SeMA J., 78 (2021), 401–414. https://doi.org/10.1007/s40324-021-00240-7 doi: 10.1007/s40324-021-00240-7
[30]	N. B. Sadabadi, F. Maheri, Fuzzy fractional conformable Laplace transforms, Honam Math. J., 43 (2021), 359–371. https://doi.org/10.5831/HMJ.2021.43.2.359 doi: 10.5831/HMJ.2021.43.2.359
[31]	M. Bataineh, M. Alaroud, S. Al-Omari, P. Agarwal, Series representations for uncertain fractional IVPs in the fuzzy conformable fractional sense, Entropy, 23 (2021), 1646. https://doi.org/10.3390/e23121646 doi: 10.3390/e23121646
[32]	Z. Al-Zhour, A. El-Ajou, M. Oqielat, O. Al-Oqily, S. Salem, M. Imran, Effective approach to construct series solutions for uncertain fractional differential equations, Fuzzy Information and Engineering, 14 (2022), 182–211. http://doi.org/10.1080/16168658.2022.2119041 doi: 10.1080/16168658.2022.2119041
[33]	T. Eriqat, A. El-Ajou, M. N. Oqielat, Z. Al-Zhour, S. Momani, A new attractive analytic approach for solutions of linear and nonlinear neutral fractional pantograph equations, Chaos Solition. Fract., 138 (2020), 109957. http://doi.org/10.1016/j.chaos.2020.109957 doi: 10.1016/j.chaos.2020.109957
[34]	M. N. Oqielat, T. Eriqat, Z. Al-Zhour, O. Ogilat, A. El-Ajou, I. Hashim, Construction of fractional series solutions to nonlinear fractional reaction–diffusion for bacteria growth model via Laplace residual power series method, Int. J. Dynam. Control, 11 (2023), 520–527. https://doi.org/10.1007/s40435-022-01001-8 doi: 10.1007/s40435-022-01001-8
[35]	M. Oqielat, T. Eriqat, Z. Al-Zhour, A. El-Ajou, S. Momani, Numerical solutions of time-fractional nonlinear water wave partial differential equation via Caputo fractional derivative: An effective analytical method and some applications, Appl. Comput. Math., 21 (2022), 207–222. http://doi.org/10.30546/1683-6154.21.2.2022.207 doi: 10.30546/1683-6154.21.2.2022.207
[36]	T. Eriqat, M. N. Oqielat, Z. Al-Zhour, G. S. Khammash, A. El-Ajou, H. Alrabaiah, Exact and numerical solutions of higher-order fractional partial differential equations: A new analytical method and some applications, Pramana, 96 (2022), 207. https://doi.org/10.1007/s12043-022-02446-4 doi: 10.1007/s12043-022-02446-4
[37]	A. El-Ajou, Z. Al-Zhour, A vector series solution for a class of hyperbolic system of Caputo-time-fractional partial differential equations with variable coefficients, Front. Phys., 9 (2021), 267. https://doi.org/10.3389/fphy.2021.525250 doi: 10.3389/fphy.2021.525250
[38]	M. N. Oqielat, T. Eriqat, O. Ogilat, A. El-Ajou, S. E. Alhazmi, S. Al-Omari, Laplace-residual power series method for solving time-fractional reaction-diffusion model, Fractal Fract., 7 (2023), 309. https://doi.org/10.3390/fractalfract7040309 doi: 10.3390/fractalfract7040309
[39]	A. El-Ajou, Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach, Eur. Phys. J. Plus, 136 (2021), 229. https://doi.org/10.1140/epjp/s13360-020-01061-9 doi: 10.1140/epjp/s13360-020-01061-9
[40]	T. Eriqat, M. N. Oqielat, Z. Al-Zhour, A. El-Ajou, A. S. Bataineh, Revisited Fisher's equation and logistic system model: A new fractional approach and some modifications, Int. J. Dynam. Control, 11 (2023), 555–563. https://doi.org/10.1007/s40435-022-01020-5 doi: 10.1007/s40435-022-01020-5
[41]	M. N. Oqielat, A. El-Ajou, Z. Al-Zhour, T. Eriqat, M. Al-Smadi, A new approach to solving fuzzy quadratic Riccati differential equations, Int. J. Fuzzy Log. Inte., 22 (2022), 23–47. http://doi.org/10.5391/IJFIS.2022.22.1.23 doi: 10.5391/IJFIS.2022.22.1.23
[42]	M. N. Oqielat, T. Eriqat, O. Ogilat, Z. Odibat, Z. Al-Zhour, I. Hashim, Approximate solutions of fuzzy fractional population dynamics model, Eur. Phys. J. Plus, 137 (2022), 982. https://doi.org/10.1140/epjp/s13360-022-03188-3 doi: 10.1140/epjp/s13360-022-03188-3
[43]	A. Qazza, R. Saadeh, E. Salah, Solving fractional partial differential equations via a new scheme, AIMS Mathematics, 8 (2023), 5318–5337. http://doi.org/10.3934/math.2023267 doi: 10.3934/math.2023267
[44]	R. Saadeh, M. Abu-Ghuwaleh, A. Qazza, E. Kuffi, A fundamental criteria to establish general formulas of integrals, J. Appl. Math., 2022 (2022), 6049367. https://doi.org/10.1155/2022/6049367 doi: 10.1155/2022/6049367
[45]	R.Saadeh, O. Ala'yed, A. Qazza, Analytical solution of coupled Hirota-Satsuma and KdV equations, Fractal Fract., 6 (2022), 694. https://doi.org/10.3390/fractalfract6120694 doi: 10.3390/fractalfract6120694
[46]	A. Qazza, M. Abdoon, R. Saadeh, M. Berir, A new scheme for solving a fractional differential equation and a Chaotic system, Eur. J. Pure Appl. Math., 16 (2023), 1128–1139. https://doi.org/10.29020/nybg.ejpam.v16i2.4769 doi: 10.29020/nybg.ejpam.v16i2.4769
[47]	R. Saadeh, M. A. Abdoon, A. Qazza, M. Berir, A numerical solution of generalized Caputo fractional initial value problems, Fractal Fract., 7 (2023), 332. https://doi.org/10.3390/fractalfract7040332 doi: 10.3390/fractalfract7040332
[48]	E. Salah, R. Saadeh, A. Qazza, R. Hatamleh, Direct power series approach for solving nonlinear initial value problems, Axioms, 12 (2023), 111. https://doi.org/10.3390/axioms12020111 doi: 10.3390/axioms12020111
[49]	M. Abu-Ghuwaleh, R. Saadeh, A. Qazza, General master theorems of integrals with applications, Mathematics, 10 (2022), 3547. https://doi.org/10.3390/math10193547 doi: 10.3390/math10193547
[50]	A. Qazza, R. Hatamleh, The existence of a solution for semi-linear abstract differential equations with infinite B-chains of the characteristic sheaf, Int. J. Appl. Math., 31 (2018), 611–620. https://doi.org/10.12732/ijam.v31i5.7 doi: 10.12732/ijam.v31i5.7
[51]	A. Amourah, A. Alsoboh, O. Ogilat, G. M. Gharib, R. Saadeh, A. A. Soudi, A generalization of gegenbauer polynomials and Bi-univalent functions, Axioms, 12 (2023), 128. http://doi.org/10.3390/axioms12020128 doi: 10.3390/axioms12020128
[52]	J. Nieto, R. Rodriguez-Lopez, M. Villanueva-Pesqueira, Exact solution to the periodic boundary value problem for a first-order linear fuzzy differential equation with impulses, Fuzzy Optim. Decis. Making, 10 (2011), 323–339. https://doi.org/10.1007/s10700-011-9108-3 doi: 10.1007/s10700-011-9108-3
[53]	A. El-Ajou, O. A. Arqub, Z. A. Zhour, S. Momani, New results on fractional power series: theories and applications, Entropy, 15 (2023), 5305–5323. http://doi.org/10.3390/e15125305 doi: 10.3390/e15125305

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