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This research paper introduces a novel fractional Caputo-type simultaneous method for finding all simple and multiple roots of polynomial equations. Without any additional polynomial and derivative evaluations using suitable correction, the order of convergence of the basic Aberth–Ehrlich simultaneous method has been increased from three to [Formula: see text]. In terms of accuracy, residual graph, computational efficiency and computation CPU time, the newly proposed families of simultaneous methods outperforms existing methods in numerical applications.

A new inverse family of the iterative method is interrogated in the present article for simultaneously estimating all distinct and multiple roots of nonlinear polynomial equations. Convergence analysis proves that the order of convergence of the newly constructed family of methods is two. The computer algebra systems CAS-Mathematica is used to determine the lower bound of convergence order, which justifies the local convergence of the newly developed method. Some nonlinear models from physics, chemistry, and engineering sciences are considered to demonstrate the performance and efficiency of the newly constructed family of inverse simultaneous methods in comparison to classical methods in the literature. The computational time in seconds and residual error graph of the inverse simultaneous methods are also presented to elaborate their convergence behavior.

In this work, we construct a new family of inverse iterative numerical technique for extracting all roots of nonlinear equation simultaneously. Convergence analysis verifies that the proposed family of methods has local 10th-order convergence. Among the test models investigated are blood rheology, a fractional nonlinear equation model, fluid permeability in biogels, and beam localization models. In comparison to other methods, the family of inverse simultaneous iterative techniques gets initial estimations to exact roots within a given tolerance while using less function evaluations in each iterative step. Numerical results, basins of attraction for fractional nonlinear equation, residual graphs are presented in detail for the simultaneous iterative techniques. The newly developed simultaneous iterative techniques were thoroughly investigated and proven to be efficient, robust, and authentic in their domain.

This paper develops optimal family of fourth-order iterative techniques in order to find a single root and to generalize them for simultaneous finding of all roots of polynomial equation. Convergence study reveals that for single root finding methods, its optimal convergence order is 4, while for simultaneous methods, it is 12. Computational cost and numerical illustrations demonstrate that the newly developed family of methods outperformed the previous methods available in the literature.

We introduce here a new two-step derivate-free inverse simultaneous iterative method for estimating all roots of nonlinear equation. It is proved that convergence order of the newly constructed method is four. Lower bound of the convergence order is determined using Mathematica and verified with theoretical local convergence order of the method introduced. Some nonlinear models which are taken from physical and engineering sciences as numerical test examples to demonstrate the performance and efficiency of the newly constructed modified inverse simultaneous methods as compared to classical methods existing in literature are presented. Dynamical planes and residual graphs are drawn using MATLAB to elaborate efficiency, robustness, and authentication in its domain.

In this article, we construct an optimal family of iterative methods for finding the single root and then extend this family for determining all the distinct as well as multiple roots of single-variable nonlinear equations simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in the case of single root finding methods and 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The computational cost, basins of attraction, efficiency, log of residual, and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in the literature.

In this article, we present a new derivative free simultaneous method for determining in particular all the distinct roots of polynomial equation. A modification in Weierstrass correction is used to construct tenth order derivative free simultaneous method, using only four function evaluation per cycle. Convergence analysis proved that the order of convergence of derivative free simultaneous method is ten and has a better computational efficiency as compared to other simultaneous methods in the literature. Numerical test examples are provided to demonstrate the performance and efficiency of the newly constructed derivative free simultaneous method.

Numerous real-world applications can be solved using the broadly adopted notions of intuitionistic fuzzy sets, Pythagorean fuzzy sets, and q-rung orthopair fuzzy sets. These theories, however, have their own restrictions in terms of membership and non-membership levels. Because it utilizes benchmark or control parameters relating to membership and non-membership levels, this theory is particularly valuable for modeling uncertainty in real-world problems. We propose the unique concept of linear Diophantine fuzzy set with benchmark parameters to overcome these restrictions. Different numerical, analytical, and semi-analytical techniques are used to solve linear systems of equations with several fuzzy numbers, such as intuitionistic fuzzy number, triangular fuzzy number, bipolar fuzzy number, trapezoidal fuzzy number, and hexagon fuzzy number. The purpose of this research is to solve a fuzzy linear system of equations with the most generalized fuzzy number, such as Triangular linear Diophantine fuzzy number, using an analytical technique called Homotopy Perturbation Method. The linear systems co-efficient are crisp when the right hand side vector is a triangular linear Diophantine fuzzy number. A numerical test examples demonstrates how our newly improved analytical technique surpasses other existing methods in terms of accuracy and CPU time. The triangular linear Diophantine fuzzy systems of equations’ strong and weak visual representations are explored.

Some complex problems in science and engineering are modeled using fuzzy differential equations. Many fuzzy differential equations cannot be solved by using exact techniques because of the complexity of the problems mentioned. We utilize analytical techniques to solve a system of fuzzy differential equations because they are simple to use and frequently result in closed-form solutions. The Generalized Modified Adomian Decomposition Method is developed in this article to compute the analytical solution to the linear system of intuitionistic triangular fuzzy initial value problems. The starting values in this case are thought of as intuitionistic triangular fuzzy numbers. Engineering examples, such as the Brine Tanks Problem, are used to demonstrate the proposed approach and show how the series solution converges to the exact solution in closed form or in series. The corresponding graphs at different levels of uncertainty show the example’s numerical outcomes. The graphical representations further demonstrate the effectiveness and accuracy of the proposed method in comparison to Taylor’s approaches and the classical Decomposition method.