Purpose: This work presents a new two-step iterative technique for solving absolute value equations. The developed technique is valuable and effective for solving the absolute value equation. Various examples are given to demonstrate the accuracy and efficacy of the suggested technique.Design/methodology/approach: In this paper, we present a new two-step iterative technique for solving absolute value equations. This technique is very straightforward, and due to the simplicity of this approach, it can be used to solve large systems with great effectiveness. Moreover, under certain assumptions, we examine the convergence of the proposed method using various theorems. Numerical outcomes are conducted to present the feasibility of the proposed technique.Findings: This paper gives numerical experiments on how to solve a system of absolute value equations.Originality/value: Nowadays, two-step approaches are very popular for solving equations (1). For solving AVEs, Liu in Shams (2021), Ning and Zhou (2015) demonstrated two-step iterative approaches. Moosaei et al. (2015) introduced a novel approach that utilizes a generalized Newton’s approach and Simpson’s rule to solve AVEs. Zainali and Lotfi (2018) presented a two-step Newton technique for AVEs that converges linearly. Feng and Liu (2016) have proposed minimization approaches for AVEs and presented their convergence under specific circumstances. Khan et al. (2023), suggested a nonlinear CSCS-like technique and a Picard-CSCS approach. Based on the benefits and drawbacks of the previously mentioned methods, we will provide a two-step iterative approach to efficiently solve equation (1). The numerical results show that our proposed technique converges rapidly and provides a more accurate solution.

This study introduces novel algebraic techniques within the framework of complex Fermatean fuzzy sets (CFFSs) by incorporating confidence levels, presenting a suite of operators tailored for advanced decision-making. Specifically, the confidence complex Fermatean fuzzy weighted geometric (CCFFWG) operator, the confidence complex Fermatean fuzzy ordered weighted geometric (CCFFOWG) operator, and the confidence complex Fermatean fuzzy hybrid geometric (CCFFHG) operator are developed to address multi-attribute group decision-making (MCGDM) challenges. These methodologies are designed to enhance decision-making in scenarios where decision-makers provide asymmetric or imprecise information, often encountered in environmental and industrial contexts. To validate the applicability of the proposed approach, a practical case study involving the selection of an optimal fire extinguisher from several alternatives is conducted. The performance of the newly developed operators is benchmarked against established methods from prior studies, with results demonstrating superior decision outcomes in terms of precision and reliability. By embedding confidence levels into complex Fermatean fuzzy operations, the proposed techniques offer greater robustness in managing uncertainty and variability across multiple attributes. These findings suggest that the advanced algebraic framework contributes significantly to improving decision quality in complex group decision-making environments.

In this study, we propose a two-step iterative procedure for solving absolute value equations. The method includes Simpson's Three-Eighths formula with five points as a corrector step and generalized Newton's approach as a predictor step. For solving large systems, this method is very effective because it is very simple. Moreover, we show the convergence analysis under certain conditions using different theorems. We conducted numerical experiments to examine the efficiency of the presented technique.

In this study, we examine the two-step technique for solving absolute value equations. In this approach, we utilize the generalized Newton approach as a predictor step. The Weddle’s formula is used as a corrector step. Furthermore, we demonstrate the convergence analysis of the suggested technique using different theorems under certain assumptions. Numerical experiments are used to demonstrate the effectiveness and feasibility of the suggested approach. Finally, we show the conclusion of the proposed technique.

In this study, we introduce the Laplace homotopy perturbation method (LHPM), based on Caputo fractional derivatives (CFDs), and demonstrate its application to approximate solutions of the fractional diffusion equation and the Korteweg–de Vries fractional‐order equation. This method’s benefit is that it can produce solutions without requiring limiting assumptions or discretisation. The usefulness and strength of the method are illustrated with four examples. The exact and approximate solutions were compared using four tables and 2D and 3D graphs. The results’ robust convergence attests to the suggested method’s dependability and effectiveness in solving fractional partial differential equations (FPDEs). Fractional diffusion equations are used to simulate anomalous transport in biological tissues, groundwater flow, porous media and finance. Nonlinear wave propagation in shallow water, plasmas, optical fibres and fluid dynamics is described by the fractional Korteweg–de Vries equation. There are many applications of the LHPM in solving complex linear and nonlinear problems in science and engineering. It works especially well with FPDEs, which are frequently found in fluid dynamics, heat transfer and viscoelastic materials. Furthermore, LHPM is used in control theory to model dynamic systems, in electrical engineering to analyse circuits with fractional elements and in biological models to describe population dynamics and the spread of disease. It is an effective tool for researching real‐world problems where exact solutions are hard to find, as it combines the Laplace transform with perturbation techniques to obtain accurate analytical approximations with rapid convergence.