A neural network-based Newton–Cotes method for solving neutrosophic fuzzy integro-differential equations

Hamzeh Zureigat; Belal Batiha; Hamzeh Zureigat; Belal Batiha

doi:10.3934/math.20261065

AIMS Mathematics

2026, Volume 11, Issue 8: 26568-26589. doi: 10.3934/math.20261065

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Research article

A neural network-based Newton–Cotes method for solving neutrosophic fuzzy integro-differential equations

Hamzeh Zureigat ^,,
Belal Batiha

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

Received: 25 April 2026 Revised: 14 July 2026 Published: 25 August 2026
MSC : 30E10, 34K05

In this article, a modified computational approach based on Newton-Cotes methods and an artificial neural network with positive coefficients is reformulated, presented, and implemented for the first time in the literature to solve the neutrosophic fuzzy integro-differential equations (NFFIDEs), where the variables and parameters are considered as fuzzy neutrosophic numbers. The neutrosophic triangular number (TNFN) is used for expressing neutrosophic fuzzy aspects. Applying fuzzification to the deterministic α-cut, β-cut, and γ-cut solutions results in a neutrosophic numerical solution based on artificial neural networks. The main aim of using neural networks was to address the neutrosophic-fuzzy aspects of NFFIDEs. A numerical experiment is provided to demonstrate the developed approach. The findings are consistent with theoretical predictions and highlight the improved accuracy and efficiency achieved by combining artificial neural networks with the Newton–Cotes method for solving NFFIDEs, supporting its applicability in medicine, engineering, and physics.
- nutrosophic fuzzy numbers,
- fuzzy neutrosophic integro-differential equations,
- Newton-Cotes methods,
- artificial neural network
Citation: Hamzeh Zureigat, Belal Batiha. A neural network-based Newton–Cotes method for solving neutrosophic fuzzy integro-differential equations[J]. AIMS Mathematics, 2026, 11(8): 26568-26589. doi: 10.3934/math.20261065

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Abstract

In this article, a modified computational approach based on Newton-Cotes methods and an artificial neural network with positive coefficients is reformulated, presented, and implemented for the first time in the literature to solve the neutrosophic fuzzy integro-differential equations (NFFIDEs), where the variables and parameters are considered as fuzzy neutrosophic numbers. The neutrosophic triangular number (TNFN) is used for expressing neutrosophic fuzzy aspects. Applying fuzzification to the deterministic α-cut, β-cut, and γ-cut solutions results in a neutrosophic numerical solution based on artificial neural networks. The main aim of using neural networks was to address the neutrosophic-fuzzy aspects of NFFIDEs. A numerical experiment is provided to demonstrate the developed approach. The findings are consistent with theoretical predictions and highlight the improved accuracy and efficiency achieved by combining artificial neural networks with the Newton–Cotes method for solving NFFIDEs, supporting its applicability in medicine, engineering, and physics.

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