Cameron University is a public university in Lawton, Oklahoma. It offers more than 50 degrees through both undergraduate and graduate programs. The degree programs emphasize the liberal arts, science and technology, and graduate and professional studies. It was founded in 1908, soon after Oklahoma was admitted as a state, as one of six agricultural high schools in the largely rural region.
The applicability of a highly efficient sixth-order convergent method, originally proposed by Kansal et al., is extended in this study to a Banach space setting. The initial development of this method relied upon Taylor series expansions in Rn and the assumption that the nonlinear operator is sufficiently differentiable. This vague condition implies the existence of high-order derivatives that are not actually utilized by the algorithm. This study transcends these limitations by establishing convergence based solely on generalized continuity conditions of the first Fr & eacute;chet derivative. By dispensing with these strong smoothness requirements, the domain of applicability is significantly widened. We derive computable radii for the ball of convergence and establish error bounds under local analysis. Furthermore, a rigorous semi-local convergence analysis is presented, a feature previously absent in the literature for this specific scheme, utilizing a majorizing sequence technique to guarantee the existence and uniqueness of the solution. The theoretical results are validated through numerical experiments, which demonstrate that the method converges even when the standard sufficiently differentiable conditions are violated.
This study introduces an optimal fourth-order iterative method derived by combining two established methods, resulting in enhanced convergence when solving nonlinear equations. Through rigorous convergence analysis using both Taylor expansion and the Banach space framework, the fourth-order optimality condition is verified. We demonstrate the superior efficiency and stability of this new method compared to traditional alternatives. Numerical experiments confirm its effectiveness, showing a reduction in the average number of iterations and computational time. Visual analysis with polynomiographs confirms the method's robustness, focusing on convergence area index, iteration count, computational time, fractal dimension, and Wada measure of basins. These findings underscore the potential of this optimal method for tackling complex nonlinear problems in various scientific and engineering fields. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http:// creativecommons.org/licenses/by-nc/4.0/).
This study introduces a class of zero-finding numerical methods that integrates the Daftardar-Jafari decomposition approach with the midpoint quadrature rule. This hybrid formulation enhances both stability and accuracy by coupling analytical decomposition with a balanced numerical integration scheme. A detailed convergence analysis-via Taylor and semilocal frameworks-demonstrates third- and fourth-order convergence, achieving up to average absolute error across benchmark nonlinear problems. The stability of the proposed solvers is visualized through polynomiography, confirming robust attraction basins compared to classical Newton-type methods. Extensive simulations on engineering and physical models (including fluid dynamics and boundary value problems) reveal faster convergence and improved efficiency indices. The paper concludes with insights on extending the method to multivariable systems and potential future developments.
We develop a class of fifth order methods introduced by Ali Zein (2023). There are many efficient fifth order methods that are special cases of this class. We present the method in a more abstract setting of a Banach space. The semilocal convergence is discussed first, and using the semilocal analysis, we obtain a ball containing the solution. The local convergence analysis does not depend on the Taylor series, and we relax the differentiability conditions on the function involved. Our assumptions for obtaining the convergence order are independent of the solution; earlier studies use assumptions involving the solution for local convergence analysis. We considered several numerical examples in chemical and physical sciences to analyze the behavior of the method. The dynamics of the method are studied.
This study examines the convergence behaviour of a fourth-order Newton-type iterative method for approximating locally unique solutions of nonlinear systems in Banach spaces. Although the method employs only first-order derivative information, existing convergence analyses frequently require assumptions on higher-order derivatives, which restrict its practical applicability. To address this issue, we establish both local and semilocal convergence results using conditions formulated solely in terms of the first derivative. Numerical experiments are presented to validate and demonstrate the theoretical findings.
