Solving intuitionistic fuzzy integro-differential equations using artificial neural networks and newton-cotes methods

Hamzeh Zureigat; Murad Algazo; Hamzeh Zureigat; Murad Algazo

doi:10.3934/math.2026387

AIMS Mathematics

2026, Volume 11, Issue 4: 9347-9364. doi: 10.3934/math.2026387

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Research article

Solving intuitionistic fuzzy integro-differential equations using artificial neural networks and newton-cotes methods

Hamzeh Zureigat ^,,
Murad Algazo

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

Received: 25 December 2025 Revised: 12 February 2026 Accepted: 05 March 2026 Published: 07 April 2026
MSC : 30E10, 34K05

In this paper, a modified method based on artificial neural network and Newton-Cotes methods with positive coefficients was developed and applied for the first time in the literature to solve the intuitionistic fuzzy integro-differential equations (IFFIDEs), where the parameters and variables are considered intuitionistic fuzzy numbers. The triangular intuitionistic fuzzy number (TIFN) was used for expressing intuitionistic fuzzy variables and parameters. The fuzzification of the deterministic r-cut and β-cut solutions leads to the artificial neural intuitionistic numerical solution. The main reason for using neural networks is their applicability in handling the intuitionistic fuzzy variables and parameters of IFFIDEs. A numerical example was given to demonstrate the proposed method. The results agree with the theoretical prediction and highlight how the combination of artificial neural networks and Newton-Cotes method enhances the accuracy and efficiency of solving IFFIDEs, making our proposed method valuable for application in fields such as engineering, medicine, and physics.
Citation: Hamzeh Zureigat, Murad Algazo. Solving intuitionistic fuzzy integro-differential equations using artificial neural networks and newton-cotes methods[J]. AIMS Mathematics, 2026, 11(4): 9347-9364. doi: 10.3934/math.2026387

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Abstract

In this paper, a modified method based on artificial neural network and Newton-Cotes methods with positive coefficients was developed and applied for the first time in the literature to solve the intuitionistic fuzzy integro-differential equations (IFFIDEs), where the parameters and variables are considered intuitionistic fuzzy numbers. The triangular intuitionistic fuzzy number (TIFN) was used for expressing intuitionistic fuzzy variables and parameters. The fuzzification of the deterministic r-cut and β-cut solutions leads to the artificial neural intuitionistic numerical solution. The main reason for using neural networks is their applicability in handling the intuitionistic fuzzy variables and parameters of IFFIDEs. A numerical example was given to demonstrate the proposed method. The results agree with the theoretical prediction and highlight how the combination of artificial neural networks and Newton-Cotes method enhances the accuracy and efficiency of solving IFFIDEs, making our proposed method valuable for application in fields such as engineering, medicine, and physics.

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