Solving Volterra fuzzy integro-differential equations using fuzzy Elzaki transform homotopy perturbation method

Hamzeh Zureigat; Murad Algazo; Hamzeh Zureigat; Murad Algazo

doi:10.3934/math.20251314

AIMS Mathematics

2025, Volume 10, Issue 12: 29901-29926. doi: 10.3934/math.20251314

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Research article

Solving Volterra fuzzy integro-differential equations using fuzzy Elzaki transform homotopy perturbation method

Hamzeh Zureigat ^,,
Murad Algazo

Department of Mathematics, Faculty of Science and Technology, Jadara University, Irbid 21110, Jordan

Received: 03 October 2025 Revised: 17 November 2025 Accepted: 21 November 2025 Published: 18 December 2025
MSC : 30E10, 34K05

Fuzzy integro-differential equations are used for modeling real-life phenomena that involve uncertain (fuzzy) parameters or variables. The combination of the fuzzy Elzaki transform and homotopy perturbation method provides a powerful hybrid technique for solving fuzzy integro-differential equations. Therefore, the aim of this paper is to modify and apply a new hybrid method called fuzzy Elzaki transform homotopy perturbation method for the first time in literature to solve fuzzy integro-differential equations. In particular, the fuzzy Elzaki transform homotopy perturbation method is developed and applied for solving linear and non-linear second-kind fuzzy Volterra integro-differential equations, and non-linear second kind fuzzy mixed Fredholm- Volterra integro-differential equations. Finally, several examples are presented to show that the fuzzy Elzaki transform homotopy perturbation method is efficient for solving wide types of fuzzy integro-differential equations with high accuracy. The novelty of this work lies in its ease of use and its high efficiency, which allows mathematicians to obtain reliable results under fuzzy Hukuhara differentiability aspects in a short time.
Citation: Hamzeh Zureigat, Murad Algazo. Solving Volterra fuzzy integro-differential equations using fuzzy Elzaki transform homotopy perturbation method[J]. AIMS Mathematics, 2025, 10(12): 29901-29926. doi: 10.3934/math.20251314

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Abstract

Fuzzy integro-differential equations are used for modeling real-life phenomena that involve uncertain (fuzzy) parameters or variables. The combination of the fuzzy Elzaki transform and homotopy perturbation method provides a powerful hybrid technique for solving fuzzy integro-differential equations. Therefore, the aim of this paper is to modify and apply a new hybrid method called fuzzy Elzaki transform homotopy perturbation method for the first time in literature to solve fuzzy integro-differential equations. In particular, the fuzzy Elzaki transform homotopy perturbation method is developed and applied for solving linear and non-linear second-kind fuzzy Volterra integro-differential equations, and non-linear second kind fuzzy mixed Fredholm- Volterra integro-differential equations. Finally, several examples are presented to show that the fuzzy Elzaki transform homotopy perturbation method is efficient for solving wide types of fuzzy integro-differential equations with high accuracy. The novelty of this work lies in its ease of use and its high efficiency, which allows mathematicians to obtain reliable results under fuzzy Hukuhara differentiability aspects in a short time.

References

[1]	Y. N. Grigoriev, N. H. Ibragimov, V. F. Kovalev, S. V. Meleshko, Symmetries of integro-differential equations, Dordrecht: Springer, 2010. https://doi.org/10.1007/978-90-481-3797-8
[2]	V. Volterra, Theory of functionals and of integral and integro-differential equations, New York: Dover, 1959.
[3]	D. S. Naidu, Optimal control systems, Boca Raton: CRC Press, 2003.
[4]	V. Lakshmikantham, M. R. M. Rao, Theory of integro-differential equations, New York: CRC Press, 1995.
[5]	I. Fredholm, Sur une classe d'équations fonctionnelles, Acta Math., 27 (1903), 365–390. https://doi.org/10.1007/BF02421317 doi: 10.1007/BF02421317
[6]	T. M. Elzaki, the new integral transform Elzaki transform, Glob. J. Pure Appl. Math., 7 (2011), 57–64.
[7]	J. H. He, Homotopy perturbation technique, Comput. Methods. Appl. Mech. Eng., 178 (1999), 257–262. https://doi.org/10.1016/S0045-7825(99)00018-3 doi: 10.1016/S0045-7825(99)00018-3
[8]	J. H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput., 156 (2004), 527–539. https://doi.org/10.1016/j.amc.2003.08.008 doi: 10.1016/j.amc.2003.08.008
[9]	S. Kambalimath, P. C. Deka, A basic review of fuzzy logic applications in hydrology and water resources, Appl. Water Sci., 10 (2020), 191. https://doi.org/10.1007/s13201-020-01276-2 doi: 10.1007/s13201-020-01276-2
[10]	D. T. Muhamediyeva, Approaches to the numerical solving of fuzzy differential equations, Int. J. Res. Eng. Technol., 3 (2014), 335–342. https://doi.org/10.15623/ijret.2014.0307057 doi: 10.15623/ijret.2014.0307057
[11]	F. Baig, M. S. Khan, Y. Noor, M. Imran, Design model of fuzzy logic medical diagnosis control system, Int. J. Comput. Sci. Eng., 3 (2011), 2093–2108.
[12]	K. Nemati, M. Matinfar, An implicit method for fuzzy parabolic partial differential equations, The J. Nonlinear Sci. Appl., 1 (2008), 61–71. https://doi.org/10.22436/jnsa.001.02.02 doi: 10.22436/jnsa.001.02.02
[13]	H. T. Nguyen, C. Walker, E. A. Walker, A first course in fuzzy logic, New York: Chapman and Hall/CRC, 2018. https://doi.org/10.1201/9780429505546
[14]	S. P. Mondal, T. K. Roy, System of differential equation with initial value as triangular intuitionistic fuzzy number and its application, Int. J. Appl. Comput. Math., 1 (2015), 449–474. https://doi.org/10.1007/s40819-015-0026-x doi: 10.1007/s40819-015-0026-x
[15]	Q. Zhou, H. Li, L. Wang, R. Lu, Prescribed performance observer-based adaptive fuzzy control for nonstrict-feedback stochastic nonlinear systems, IEEE Trans. Syst. Man Cybern. Syst., 48 (2017), 1747–1758. https://doi.org/10.1109/TSMC.2017.2738155 doi: 10.1109/TSMC.2017.2738155
[16]	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
[17]	L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
[18]	H. V. Long, N. T. K. Son, N. V. Hoa, Fuzzy fractional partial differential equations in partially ordered metric spaces, Iran. J. Fuzzy Syst., 14 (2017), 107–126. https://doi.org/10.22111/ijfs.2017.3136 doi: 10.22111/ijfs.2017.3136
[19]	O. Akın, O. Oruc, A prey and predator model with fuzzy initial values, Hacet. J. Math. Stat., 41 (2012), 387–395.
[20]	A. Torres, J. J. Nieto, Fuzzy logic in medicine and bioinformatics, Biomed Res. Int., 2006 (2006), 091908. https://doi.org/10.1155/JBB/2006/91908 doi: 10.1155/JBB/2006/91908
[21]	P. Guttorp, Fuzzy mathematical models in engineering and management science, Technometrics, 32 (1990), 238. https://doi.org/10.1080/00401706.1990.10484661 doi: 10.1080/00401706.1990.10484661
[22]	S. Chakraverty, S. Tapaswini, D. Behera, Fuzzy differential equations and applications for engineers and scientists, Boca Raton: CRC Press, 2016. https://doi.org/10.1201/9781315372853
[23]	O. Abu Arqub, R. Mezghiche, B. Maayah, Fuzzy M-fractional integrodifferential models: theoretical existence and uniqueness results, and approximate solutions utilizing the Hilbert reproducing kernel algorithm, Front. Phys., 11 (2023), 1252919. https://doi.org/10.3389/fphy.2023.1252919 doi: 10.3389/fphy.2023.1252919
[24]	R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets Syst., 18 (1986), 31–43. https://doi.org/10.1016/0165-0114(86)90026-6 doi: 10.1016/0165-0114(86)90026-6
[25]	D. Dubois, H. Prade, Systems of linear fuzzy constraints, Fuzzy Sets Syst., 3 (1980), 37–48. https://doi.org/10.1016/0165-0114(80)90004-4 doi: 10.1016/0165-0114(80)90004-4
[26]	M. L. Puri, D. A. Ralescu, L. Zadeh, Fuzzy random variables, In: Readings in fuzzy sets for intelligent systems, 1993,265–271. https://doi.org/10.1016/b978-1-4832-1450-4.50029-8
[27]	M. Hukuhara, Integration des applications mesurables dont la valeur est un compact convexe, Funkcial. Ekvac., 10 (1967), 205–223.
[28]	S. S. Mansouri, N. Ahmady, A numerical method for solving nth-order fuzzy differential equation by using characterization theorem, Commun. Nonlinear Sci., 2012 (2012), cna-00054.
[29]	L. Stefanini, L. Sorini, M. L. Guerra, Parametric representation of fuzzy numbers and application to fuzzy calculus, Fuzzy Sets Syst., 157 (2006), 2423–2455. https://doi.org/10.1016/j.fss.2006.02.002 doi: 10.1016/j.fss.2006.02.002
[30]	M. Friedman, M. Ma, A. Kandel, Numerical solutions of fuzzy differential and integral equations, Fuzzy Sets Syst., 106 (1999), 35–48. https://doi.org/10.1016/s0165-0114(98)00355-8 doi: 10.1016/s0165-0114(98)00355-8
[31]	H. C. Wu, The improper fuzzy Riemann integral and its numerical integration, Inf. Sci., 111 (1998), 109–137. https://doi.org/10.1016/s0020-0255(98)00016-4 doi: 10.1016/s0020-0255(98)00016-4
[32]	T. M. Elzaki, S. M. Elzaki, On the Tarig Transform and ordinary differential equation with variable coefficients, Elixir Appl. Math., 38 (2011), 4250–4252.
[33]	H. Kim, The time shifting theorem and the convolution for Elzaki transform, Int. J. Pure Appl. Math., 87 (2013), 261–271. https://doi.org/10.12732/ijpam.v87i2.6 doi: 10.12732/ijpam.v87i2.6
[34]	A. Ghorbani, Beyond Adomian polynomials: He polynomials, Chaos Soliton Fract., 39 (2009), 1486–1492. https://doi.org/10.1016/j.chaos.2007.06.034 doi: 10.1016/j.chaos.2007.06.034
[35]	H. M. Ahmed, An advanced approach for numerical solution of a class of Fredholm-Volterra integro-differential equations with mixed boundary conditions, Int. J. Mod. Phys. C, 2025. https://doi.org/10.1142/S0129183125501542 doi: 10.1142/S0129183125501542
[36]	S. M. Hosseini, S. Shahmorad, Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method, Appl. Math. Model., 29 (2005), 1005–1021. https://doi.org/10.1016/j.apm.2005.02.003 doi: 10.1016/j.apm.2005.02.003
[37]	S. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004), 499–513. https://doi.org/10.1016/S0096-3003(02)00790-7 doi: 10.1016/S0096-3003(02)00790-7
[38]	O. Nave, V. Gol'dshtein, A combination of two semi-analytical methods called "singular perturbed homotopy analysis method (SPHAM)" applied to combustion of spray fuel droplets, Cogent Math., 3 (2016), 1256467. https://doi.org/10.1080/23311835.2016.1256467 doi: 10.1080/23311835.2016.1256467
[39]	T. Liu, Porosity reconstruction based on Biot elastic model of porous media by homotopy perturbation method, Chaos Soliton Fract., 158 (2022), 112007. https://doi.org/10.1016/j.chaos.2022.112007 doi: 10.1016/j.chaos.2022.112007

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