Abbas I. Khlaif^1,2This email address is being protected from spambots. You need JavaScript enabled to view it., Osama H. Mohammed³, and Moez Feki⁴

¹Research group LAMMDA, Higher School of Sciences and Technology of Hammam Sousse, University of Sousse, Tunisia.

²Department of Mathematics and Computer Applications, College of Sciences, Al-Nahrain University, Jadriya , Baghdad, Iraq.

³Department of Mathematics and Computer Applications, College of Sciences, Al-Nahrain University, Jadriya , Baghdad, Iraq.(E-mail This email address is being protected from spambots. You need JavaScript enabled to view it.)

⁴Research group LAMMDA, Higher School of Sciences and Technology of Hammam Sousse, University of Sousse, Tunisia.

1. [1] S. H. Kilbas A.A. and TrujilloJ.J. Theory and applications of Fractional Differential Equations. 204. Elsevier, 2006, 1–523.
2. [2] I. Podlubny. Fractional differential equations, mathematics in science and engineering. 1999.
3. [3] R. Hilfer. Applications of fractional calculus in physics. World scientific, 2000.
4. [4] D. Y. Ye H. and Gao.J., (2007) “The Existence of a Positive Solution of Dα [x(t) − x(0)] = x(t)f(t, xt) )" positivity 11: 341–350. DOI: 10.1007/s11117-006-2038-6.
5. [5] K. A. Jalil AFA, (2022) “Toward solving fractional differential equations via solving ordinary differential equations." Computational and applied mathematics 41(37): 1–12. DOI: 10.1007/s40314-021-01744-8.
6. [6] e. a. Khudair AR Haddad S, (2017) “Restricted fractional differential transform for solving irrational order fractional differential equations" Chaos Solitons Fractals 101: 81–85. DOI: 10.1016/j.chaos.2017.05.026.
7. [7] A. N. Armeina D Rusyaman E, (2021) “Convergence of solution function sequences of non-homogenous fractional partial differential equation solution using homotopy analysis method (HAM)" AIP Conference Proceedings 2329(1): DOI: 10.1063/5.0042171.
8. [8] M. Arfan, I. Mahariq, K. Shah, T. Abdeljawad, G. Laouini, and P. O. Mohammed, (2022) “Numerical computations and theoretical investigations of a dynamical system with fractional order derivative" Alexandria Engineering Journal 61(3): 1982–1994. DOI: 10.1016/j.aej.2021.07.014.
9. [9] R. Amin, H. Ahmad, K. Shah, M. B. Hafeez, and W. Sumelka, (2021) “Theoretical and computational analysis of nonlinear fractional integro-differential equations via collocation method" Chaos, Solitons & Fractals 151: 111252. DOI: 10.1016/j.chaos.2021.111252.
10. [10] M. A. Almalahi, S. K. Panchal, W. Shatanawi, M. S. Abdo, K. Shah, and K. Abodayeh, (2021) “Analytical study of transmission dynamics of 2019-nCoV pandemic via fractal fractional operator" Results in physics 24: 104045. DOI: 10.1016/j.rinp.2021.104045.
11. [11] R. Amin, K. Shah, M. Asif, and I. Khan, (2020) “Efficient numerical technique for solution of delay VolterraFredholm integral equations using Haar wavelet" Heliyon 6(10): DOI: 10.1016/j.heliyon.2020.e05108.
12. [12] M. Nadeem and J.-H. He, (2022) “The homotopy perturbation method for fractional differential equations: Part 2, two-scale transform" International Journal of Numerical Methods for Heat & Fluid Flow 32(2): 559–567.
13. [13] C. Liao and H. Ye, (2009) “Existence of positive solutions of nonlinear fractional delay differential equations" Positivity 13: 601–609.
14. [14] M. L. Morgado, N. J. Ford, and P. M. Lima, (2013) “Analysis and numerical methods for fractional differential equations with delay" Journal of computational and applied mathematics 252: 159–168.
15. [15] J. Džurina, S. R. Grace, I. Jadlovská, and T. Li, (2020) “Oscillation criteria for second-order Emden–Fowler delay differential equations with a sublinear neutral term" Mathematische Nachrichten 293(5): 910–922.
16. [16] A. Jhinga and V. Daftardar-Gejji, (2019) “A new numerical method for solving fractional delay differential equations" Computational and Applied Mathematics 38: 1–18.
17. [17] S. Shen, F. Liu, J. Chen, I. Turner, and V. Anh, (2012) “Numerical techniques for the variable order time fractional diffusion equation" Applied Mathematics and Computation 218(22): 10861–10870.
18. [18] Y. Shekari, A. Tayebi, and M. H. Heydari, (2019) “A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation" Computer Methods in Applied Mechanics and Engineering 350: 154–168.
19. [19] J. Liu, X. Li, and X. Hu, (2019) “A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation" Journal of Computational Physics 384: 222– 238.
20. [20] H. Wang and X. Zheng, (2019) “Wellposedness and regularity of the variable-order time-fractional diffusion equations" Journal of Mathematical Analysis and Applications 475(2): 1778–1802.
21. [21] D. Tavares, R. Almeida, and D. F. Torres, (2016) “Caputo derivatives of fractional variable order: numerical approximations" Communications in Nonlinear Science and Numerical Simulation 35: 69–87.
22. [22] O. H. Mohammed and I. K. Abbas, (2014) “Homotopy Analysis Method for solving delay differential equations of fractional order" Mathematical Theory and Modeling 4(14):
23. [23] A. N. Talib and O. H. Mohammed, (2022) “Artificial Neural Network Technique for Solving Variable Order Fractional Integro-Differential Algebraic Equations" AlNahrain Journal of Science 25(3): 25–32.
24. [24] A. K. Farhood and O. H. Mohammed, (2023) “Homotopy perturbation method for solving time-fractional nonlinear Variable-Order Delay Partial Differential Equations" Partial Differential Equations in Applied Mathematics 7: 100513.
25. [25] M. G. Sakar, F. Uludag, and F. Erdogan, (2016) “Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method" Applied Mathematical Modelling 40(13-14): 6639–6649.
26. [26] J. Biazar and H. Ghazvini, (2009) “Convergence of the homotopy perturbation method for partial differential equations" Nonlinear Analysis: Real World Applications 10(5): 2633–2640.
27. [27] Z. Wang, L. Zou, and Y. Qin, (2017) “Piecewise homotopy analysis method and convergence analysis for formally well-posed initial value problems" Numerical Algorithms 76: 393–411.
28. [28] A. K. Farhood, O. H. Mohammed, and B. A. Taha, (2023) “Solving fractional time-delay diffusion equation with variable-order derivative based on shifted Legendre– Laguerre operational matrices" Arabian Journal of Mathematics 12(3): 529–539.
29. [29] A. K. Farhood and O. H. Mohammed, (2023) “Shifted Chebyshev operational matrices to solve the fractional time-delay diffusion equation" Partial Differential Equations in Applied Mathematics 8: 100538.
30. [30] H. Qu, Z. She, and X. Liu, (2020) “Homotopy analysis method for three types of fractional partial differential equations" Complexity 2020(1): 7232907.
31. [31] S. Liao. Beyond perturbation: introduction to the homotopy analysis method. 2003.
32. [32] L. Akinyemi, M. ¸Senol, and S. N. Huseen, (2021) “Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsov equation in dusty plasma" Advances in Difference Equations 2021(1): 45.
33. [33] M. S. Ismael, F. S. Fadhel, and A. Al-Fayadh, (2020) “Approximate Solution of Multi-Term Fractional Order Delay Differential Equations Using Homotopy Perturbation Method" Al-Nahrain Journal of Science 23(2): 60–66. DOI: 10.1063/5.0118320.
34. [34] L. A. Zadeh, (1965) “Information and control" Fuzzy sets 8(3): 338–353.
35. [35] A. K. Sahni, J. T. Pandey, R. K. Mishra, and V. Kumar, (2021) “On fuzzy proper exact sequences and fuzzy projective semi-modules over semirings" WSEAS Transactions on Mathematics 20: 700–711.
36. [36] R. Goetschel Jr and W. Voxman, (1986) “Elementary fuzzy calculus" Fuzzy sets and systems 18(1): 31–43.
37. [37] D. Dubois and H. Prade, (1982) “Towards fuzzy differential calculus part 1: Integration of fuzzy mappings" Fuzzy sets and Systems 8(1): 1–17.
38. [38] Y. Chalco-Cano and H. Roman-Flores, (2008) “On new solutions of fuzzy differential equations" Chaos, Solitons & Fractals 38(1): 112–119.
39. [39] W. Congxin and M. Ming, (1990) “On the integrals, series and integral equations of fuzzy set valued functions" Journal Of Harbin Institute Of Technology:
40. [40] F. S. Fadhel, J. H. Eidi, H. M. Wali, et al., (2021) “Contraction mapping theorem in partial fuzzy metric spaces" Journal of Applied Science and Engineering 25(2): 353–360. DOI: 10.6180/jase.202204_25(2).0020.. [41] N. R. Kareem, F. S. Fadhel, and S. Al-Nassir, (2024) “Solution Of Linear Fuzzy Stochastic Ordinary Differential Equations Using Homotopy Perturbation Method" Journal of Applied Science and Engineering 28(4): 717–730.