A higher-order iterative scheme and its integration with the Grey Wolf Optimization algorithm for solving complex nonlinear systems

Aanchal Chandel; Sonia Bhalla; Gurjeet Singh; Alicia Cordero; Juan R. Torregrosa; Aanchal Chandel; Sonia Bhalla; Gurjeet Singh; Alicia Cordero; Juan R. Torregrosa

doi:10.3934/math.20261054

AIMS Mathematics

2026, Volume 11, Issue 8: 26273-26300. doi: 10.3934/math.20261054

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Research article

A higher-order iterative scheme and its integration with the Grey Wolf Optimization algorithm for solving complex nonlinear systems

1.
Department of Mathematics, Chandigarh University, Gharuan, Mohali, 140413, Punjab, India
2.
Department of Mathematics, RIMT University, Mandi, Gobindgarh, 147301, Punjab, India
3.
Department of Mathematics, Universitat Politècnica de València, 46022, Valencia, Spain

Received: 05 June 2026 Revised: 10 August 2026 Accepted: 13 August 2026 Published: 24 August 2026
MSC : 65H05, 65G99

This study proposes a novel iterative scheme for an effective solution of systems of nonlinear equations (SoNE) that improves the order of convergence of the classical methods. Along with the proposed deterministic scheme, a hybrid computational framework is produced by combining the proposed iterative technique with the Grey Wolf Optimization (GWO) algorithm. The goal of integration is to combine the iterative method's strong local search ability with the metaheuristic optimization technique's global search ability. This will make the process more robust and less sensitive to initial guesses. To determine how reliable and efficient the proposed iterative scheme is, we look at its theoretical convergence order. Additionally, a comprehensive statistical analysis is tested to assess the efficacy of the proposed methods regarding accuracy, convergence rate, and stability. Numerical experiments are conducted on various nonlinear problems to validate the efficacy of the methodologies. To demonstrate their practical utility, the proposed methods are employed on real-world chemical engineering models characterized by nonlinear dependence among process variables. The computational results show that the proposed methods produce accurate numerical solutions with a better convergence order.
- Newton-type methods,
- metaheuristic techniques,
- convergence analysis,
- nonlinear system
Citation: Aanchal Chandel, Sonia Bhalla, Gurjeet Singh, Alicia Cordero, Juan R. Torregrosa. A higher-order iterative scheme and its integration with the Grey Wolf Optimization algorithm for solving complex nonlinear systems[J]. AIMS Mathematics, 2026, 11(8): 26273-26300. doi: 10.3934/math.20261054

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Abstract

This study proposes a novel iterative scheme for an effective solution of systems of nonlinear equations (SoNE) that improves the order of convergence of the classical methods. Along with the proposed deterministic scheme, a hybrid computational framework is produced by combining the proposed iterative technique with the Grey Wolf Optimization (GWO) algorithm. The goal of integration is to combine the iterative method's strong local search ability with the metaheuristic optimization technique's global search ability. This will make the process more robust and less sensitive to initial guesses. To determine how reliable and efficient the proposed iterative scheme is, we look at its theoretical convergence order. Additionally, a comprehensive statistical analysis is tested to assess the efficacy of the proposed methods regarding accuracy, convergence rate, and stability. Numerical experiments are conducted on various nonlinear problems to validate the efficacy of the methodologies. To demonstrate their practical utility, the proposed methods are employed on real-world chemical engineering models characterized by nonlinear dependence among process variables. The computational results show that the proposed methods produce accurate numerical solutions with a better convergence order.

References

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