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Volume 14
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Article

First Optimal Eighth-Order Families with Multivariable Scalar Weight Functions for Nonlinear Systems and Applications to Fredholm Integral and Semilinear Elliptic Problems

by
Alicia Cordero
1,
Miguel A. Leonardo Sepúlveda
2,3,*,
Juan R. Torregrosa
1,
Antmel Rodríguez Cabral
4 and
Natanael Ureña Castillo
4,5
1
Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
2
Departamento de Matemática, Universidad APEC (UNAPEC), Avenida Máximo Gómez No. 72, Santo Domingo 10107, Dominican Republic
3
Departamento de Matemática, Instituto Superior de Formación Docente Salomé Ureña (ISFODOSU), Av. Caonabo, Santo Domingo 10114, Dominican Republic
4
Escuela de Matemáticas, Universidad Autónoma de Santo Domingo (UASD), Ciudad Universitaria, Av. Alma Mater, Santo Domingo 10105, Dominican Republic
5
Ciencias Básicas y Ambientales (CBA), Instituto Tecnológico de Santo Domingo (INTEC), Santo Domingo 10602, Dominican Republic
*
Author to whom correspondence should be addressed.
Mathematics 2026, 14(12), 2114; https://doi.org/10.3390/math14122114
Submission received: 24 May 2026 / Revised: 8 June 2026 / Accepted: 11 June 2026 / Published: 13 June 2026
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations: Recent Developments, Analysis and Applications)
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Abstract

This paper presents new optimal eighth-order families with weight functions for solving nonlinear systems, obtained as a generalization of the first optimal eighth-order CTT8 method introduced by Cordero, Torregrosa and Triguero-Navarro. The proposed schemes are constructed by combining a Newton-type predictor with high-order correction steps whose weight functions are suitably chosen to preserve optimal convergence while keeping a low computational cost. To the best of our knowledge, this work introduces the first family of optimal eighth-order methods for nonlinear systems, in the sense of the Cordero–Torregrosa conjecture, developed through a weight-function technique. A complete local convergence analysis is carried out under standard smoothness assumptions, proving eighth-order convergence for nondegenerate solutions. The computational efficiency of the proposed methods is also studied and compared with several existing high-order iterative schemes. Numerical experiments on nonlinear systems of different dimensions confirm the theoretical order of convergence and show the robustness of the new families. In addition, a Fredholm integral equation is solved, followed by a semilinear elliptic Dirichlet problem, further illustrating the reliability and computational performance of the proposed weight-function-based methods.
Keywords: nonlinear systems; iterative methods; eighth-order convergence; multivariable weight functions; optimal methods; Fredholm integral equation; semilinear elliptic Dirichlet problem nonlinear systems; iterative methods; eighth-order convergence; multivariable weight functions; optimal methods; Fredholm integral equation; semilinear elliptic Dirichlet problem

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MDPI and ACS Style

Cordero, A.; Leonardo Sepúlveda, M.A.; Torregrosa, J.R.; Rodríguez Cabral, A.; Ureña Castillo, N. First Optimal Eighth-Order Families with Multivariable Scalar Weight Functions for Nonlinear Systems and Applications to Fredholm Integral and Semilinear Elliptic Problems. Mathematics 2026, 14, 2114. https://doi.org/10.3390/math14122114

AMA Style

Cordero A, Leonardo Sepúlveda MA, Torregrosa JR, Rodríguez Cabral A, Ureña Castillo N. First Optimal Eighth-Order Families with Multivariable Scalar Weight Functions for Nonlinear Systems and Applications to Fredholm Integral and Semilinear Elliptic Problems. Mathematics. 2026; 14(12):2114. https://doi.org/10.3390/math14122114

Chicago/Turabian Style

Cordero, Alicia, Miguel A. Leonardo Sepúlveda, Juan R. Torregrosa, Antmel Rodríguez Cabral, and Natanael Ureña Castillo. 2026. "First Optimal Eighth-Order Families with Multivariable Scalar Weight Functions for Nonlinear Systems and Applications to Fredholm Integral and Semilinear Elliptic Problems" Mathematics 14, no. 12: 2114. https://doi.org/10.3390/math14122114

APA Style

Cordero, A., Leonardo Sepúlveda, M. A., Torregrosa, J. R., Rodríguez Cabral, A., & Ureña Castillo, N. (2026). First Optimal Eighth-Order Families with Multivariable Scalar Weight Functions for Nonlinear Systems and Applications to Fredholm Integral and Semilinear Elliptic Problems. Mathematics, 14(12), 2114. https://doi.org/10.3390/math14122114

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