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A novel three-step simultaneous scheme for finding all distinct and multiple roots of non-linear equations are developed in this paper. Analysis of convergence demonstrates that the newly created scheme has an order of convergence of eight. A hybrid neural network-based simultaneous technique is also developed to expedite convergence. Neural network-based approaches outperform existing methods in terms of iterations, error, and CPU time, as demonstrated by the engineering applications’ results. The analysis of the considered approaches on random starting approximations reveals that the newly proposed methods are more stable and consistent than previous methods.

Nonlinear equations are important in engineering because they may simulate complicated real-world phenomena such as fluid dynamics, material stress, and electrical circuits, where linear assumptions fail. They allow engineers to more correctly estimate how a system will respond in certain conditions. The main aim of this effort is to develop an efficient higher-order simultaneous computer approach capable of computing all solutions concurrently. The convergence theorem analysis indicates that the scheme has a local convergence order of 10. Using a few engineering applications, we show that the order strategy surpasses the current approach in terms of residual error, stability, and consistency.

This study extended an existing semi-analytical technique, the Homotopy Perturbation Method, to the Block Homotopy Modified Perturbation Method by solving two [Formula: see text] crisp triangular intuitionistic fuzzy (TIF) systems of linear equations. In the original system, the coefficient matrix is considered as real crisp, while the unknown variable vector and right hand side vector are regarded as triangular intuitionistic fuzzy numbers. The Block Homotopy Modified Perturbation Method is found to be efficient and practical to solve [Formula: see text] TIF linear systems as it only requires the non-singularity of the [Formula: see text] TIF linear system's coefficient matrix, whereas the point Homotopy Perturbation Method and other classical numerical iterative methods typically require non-zero diagonal entries in the coefficient matrix. A set of theorems relevant to this study are presented and demonstrated. We solve an engineering application, i.e. a current flow circuit problem that is represented in terms of a triangular intuitionistic fuzzy environment, using the suggested method. The unknown current is then obtained as a triangle intuitionistic fuzzy number. The proposed semi-analytic method is used to solve some numerical test problems in order to validate their performance and efficiency in comparison to other existing techniques. The numerical results of the example are displayed on graphs with different degrees of uncertainty. The efficiency and accuracy of the proposed method are further demonstrated by comparisons to block Jacobi, Adomain Decomposition method, Successive Over-Relaxation method and the classical Gauss-Seidel numerical method.

This study shows the link between computer science and applied mathematics. It conducts a dynamics investigation of new root solvers using computer tools and develops a new family of single-step simple root-finding methods. The convergence order of the proposed family of iterative methods is two, according to the convergence analysis carried out using symbolic computation in the computer algebra system CAS-Maple 18. Without further evaluations of a given nonlinear function and its derivatives, a very rapid convergence rate is achieved, demonstrating the remarkable computing efficiency of the novel technique. To determine the simple roots of nonlinear equations, this paper discusses the dynamic analysis of one-parameter families using symbolic computation, computer animation, and multi-precision arithmetic. To choose the best parametric value used in iterative schemes, it implements the parametric and dynamical plane technique using CAS-MATLAB$ ^{@}R2011b. $ The dynamic evaluation of the methods is also presented utilizing basins of attraction to analyze their convergence behavior. Aside from visualizing iterative processes, this method illustrates not only iterative processes but also gives useful information regarding the convergence of the numerical scheme based on initial guessed values. Some nonlinear problems that arise in science and engineering are used to demonstrate the performance and efficiency of the newly developed method compared to the existing method in the literature.

In this study, we develop new efficient parallel techniques for solving both distinct and multiple roots of nonlinear problems at the same time. The parallel techniques represent an innovative contribution to the discipline, with local convergence of the ninth order. Theoretical research shows the rapid convergence and effectiveness of the proposed parallel schemes. To assess the suggested scheme’s stability and consistency, we look at certain biomedical engineering applications, such as osteoporosis in Chinese women, blood rheology, and differential equations. Overall, detailed analyses of convergence behavior, memory utilization, computational time, and percentage computational efficiency show that the novel parallel techniques outperform the traditional methods. The proposed methods would be more suitable for large-scale computational problems in biomedical applications due to their advantages in memory efficiency, CPU time, and error reduction.

Nonlinear equations are essential in research and engineering because they simulate complicated processes such as fluid dynamics, chemical reactions, and population growth. The development of advanced methods to address them becomes essential for scientific and applied research enhancements, as their resolution influences innovations by aiding in the proper prediction or optimization of the system. In this research, we develop a novel biparametric family of inverse parallel techniques designed to improve stability and accelerate convergence in parallel iterative algorithm. Bifurcation and chaos theory were used to find the best parameter regions that increase the parallel method’s effectiveness and stability. Our newly developed biparametric family of parallel techniques is more computationally efficient than current approaches, as evidenced by significant reductions in the number of iterations and basic operations each iterations step for solving nonlinear equations. Engineering applications examined with rough beginning data demonstrate high accuracy and superior convergence compared to existing classical parallel schemes. Analysis of global convergence further shows that the proposed methods outperform current methods in terms of error control, computational time, percentage convergence, number of basic operations per iteration, and computational order. These findings indicate broad usage potential in engineering and scientific computation.

In this article, we constructed a derivative-free family of iterative techniques for extracting simultaneously all the distinct roots of a non-linear polynomial equation. Convergence analysis is discussed to show that the proposed family of iterative method has fifth order convergence. Nonlinear test models including fractional conversion, predator-prey, chemical reactor and beam designing models are included. Also many other interesting results concerning symmetric problems with application of group symmetry are also described. The simultaneous iterative scheme is applied starting with the initial estimates to get the exact roots within the given tolerance. The proposed iterative scheme requires less function evaluations and computation time as compared to existing classical methods. Dynamical planes are exhibited in CAS-MATLAB (R2011B) to show how the simultaneous iterative approach outperforms single roots finding methods that might confine the divergence zone in terms of global convergence. Furthermore, convergence domains, namely basins of attraction that are symmetrical through fractal-like edges, are analyzed using the graphical tool. Numerical results and residual graphs are presented in detail for the simultaneous iterative method. An extensive study has been made for the newly developed simultaneous iterative scheme, which is found to be efficient, robust and authentic in its domain.

Fractional-order nonlinear equation-solving methods are crucial in engineering, where complex system modeling requires great precision and accuracy. Engineers may design more reliable mechanisms, enhance performance, and develop more accurate predictions regarding outcomes across a range of applications where these problems are effectively addressed. This research introduces a novel hybrid multiplicative calculus-based parallel method for solving complex nonlinear models in engineering. To speed up the method’s rate of convergence, we utilize a second-order multiplicative root-finding approach as a corrector in the parallel framework. Using rigorous theoretical analysis, we illustrate how the hybrid parallel technique based on multiplicative calculus achieves a remarkable convergence order of 12, indicating its effectiveness and efficiency in solving complex nonlinear equations. The intrinsic stability and consistency of the approach—when applied to nonlinear situations—are clearly indicated by the symmetry seen in the dynamical planes for various parameter values. The method’s symmetrical behavior indicates that it produces accurate findings under a range of scenarios. Using a dynamical system procedure, the ideal parameter values are systematically analyzed in order to further improve the method’s performance. Implementing the aforementioned parameter values using the parallel approach yields very reliable and consistent outcomes. The method’s effectiveness, reliability, and consistency are evaluated through the analysis of numerous nonlinear engineering problems. The analysis provides a detailed comparison with current techniques, emphasizing the benefits and potential improvements of the novel approach.

It is essential to solve nonlinear equations in engineering, where accuracy and precision are critical. In this paper, a novel family of iterative methods for finding the simple roots of nonlinear equations based on multiplicative calculus is introduced. Based on theoretical research, a novel family of simple root-finding schemes based on multiplicative calculus has been devised, with a convergence order of seven. The symmetry in the pie graph of the convergence–divergence areas demonstrates that the method is stable and consistent when dealing with nonlinear engineering problems. An extensive examination of the numerical results of the engineering applications is presented in order to assess the effectiveness, stability, and consistency of the recently established method in comparison to current methods. The analysis includes the total number of functions and derivative evaluations per iteration, elapsed time, residual errors, local computational order of convergence, and error graphs, which demonstrate our method’s better convergence behavior when compared to other approaches.

Nonlinear problems, which often arise in various scientific and engineering disciplines, typically involve nonlinear equations or functions with multiple solutions. Analytical solutions to these problems are often impossible to obtain, necessitating the use of numerical techniques. This research proposes an efficient and stable Caputo-type inverse numerical fractional scheme for simultaneously approximating all roots of nonlinear equations, with a convergence order of 2ψ+2. The scheme is applied to various nonlinear problems, utilizing dynamical analysis to determine efficient initial values for a single root-finding Caputo-type fractional scheme, which is further employed in inverse fractional parallel schemes to accelerate convergence rates. Several sets of random initial vectors demonstrate the global convergence behavior of the proposed method. The newly developed scheme outperforms existing methods in terms of accuracy, consistency, validation, computational CPU time, residual error, and stability.

The roots of non-linear equations are a major challenge in many scientific and professional fields. This problem has been approached in a number of ways, including use of the sequential Newton's method and the traditional Weierstrass simultaneous iterative scheme. To approximate all of the roots of a given nonlinear equation, sequential iterative algorithms must use a deflation strategy because rounding errors can produce inaccurate results. This study aims to develop an efficient numerical simultaneous scheme for approximating all nonlinear equations' roots of convergence order 12. The numerical outcomes of the considered engineering problems show that, in terms of accuracy, validations, error, computational CPU time, and residual error, recently developed simultaneous methods perform better than existing methods in the literature.

In scientific and engineering disciplines, vectorial problems involving systems of equations or functions with multiple variables frequently arise, often defying analytical solutions and necessitating numerical techniques. This research introduces an efficient numerical scheme capable of simultaneously approximating all roots of nonlinear equations with a convergence order of ten, specifically designed for vectorial problems. Random initial vectors are employed to assess the global convergence behavior of the proposed scheme. The newly developed method surpasses methods in the existing literature in terms of accuracy, consistency, computational CPU time, residual error, and stability. This superiority is demonstrated through numerical experiments tackling engineering problems and solving heat equations under various diffusibility parameters and boundary conditions. The findings underscore the efficacy of the proposed approach in addressing complex nonlinear systems encountered in diverse applied scenarios.

Fractional calculus can be used to fully describe numerous real-world situations in a wide range of scientific disciplines, including natural science, social science, electrical, chemical, and mechanical engineering, economics, statistics, weather forecasting, and particularly biomedical engineering. Different types of derivatives can be useful when solving various fractional calculus problems. In this study, we suggested a single step modified one parameter family of the Caputo-type fractional iterative method. Convergence analysis shows that the proposed family of methods' order of convergence is $ \vartheta +1 $ . To determine the error equation of the proposed technique, the computer algebra system CAS-Maple is employed. To illustrate the accuracy, validity, and usefulness of the proposed technique, we consider a few real-world applications from the fields of civil and chemical engineering. In terms of residual error, computational time, computational order of convergence, efficiency, and absolute error, the test examples' acquired numerical results demonstrate that the newly proposed algorithm performs better than the other classical fractional iterative scheme already existing in the literature. Using the computer program Mathematia 9.0, we compare the draw basins of attraction of the suggested fractional numerical algorithm to those of the currently used fractional iterative methods for the graphical analysis. The graphical results show how quickly the newly developed fractional method converges, confirming its supremacy to other techniques.

The aim of this article is to implement the Generalized Modified Adomian Decomposition Method to compute the semi-numerical solution of the linear system of intuitionistic fuzzy initial value problems. Here, we consider the initial values as generalized trapezoidal intuitionistic fuzzy numbers. The technique is applied to brine tanks problem and coupled mass spring systems.Theoretically, different approaches to solving a system of generalized trapezoidal intuitionistic fuzzy differential equations are discussed in this study under the presumption that the coefficients of the system of the differential equations are associated to generalized trapezoidal intuitionistic fuzzy numbers. The approximate results are compared with exact solutions which shows good efficiency. 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 existing semi-numerical methods in the literature.

A family of three-step optimal eighth-order iterative algorithm is developed in this paper in order to find single roots of nonlinear equations using the weight function technique. The newly proposed iterative methods of eight order convergence need three function evaluations and one first derivative evaluation that satisfies the Kung–Traub optimality conjecture in terms of computational cost per iteration (i.e. $ {2^{n - 1}} $ ). Furthermore, using the primary theorem that establishes the convergence order, the theoretical convergence properties of our schemes are thoroughly investigated. On several engineering applications, the performance and efficiency of our optimal iteration algorithms are examined to those of existing competitors. The new iterative schemes are more efficient than the existing methods in the literature, as illustrated by the basins of attraction, dynamical planes, efficiency, log of residual, and numerical test examples.

In this research, we suggested a numerical iterative scheme for investigating the numerical solution of fuzzy linear and nonlinear systems of equations, particularly where the linear or nonlinear system co-efficient is a crisp number and the right-hand side vector is a triangular fuzzy number. Triangular fuzzy systems of linear and nonlinear equations play a critical role in a variety of engineering, scientific challenges, mathematics, chemistry, physics, artificial intelligence, biology, medical, economics, finance, statistics, machine and deep learning, computer science, robotics and smart cars, programming, in the military and engineering industries, linear and nonlinear programming problems and traffic flow problems. In biomedical engineering, fluid flow problems, and differential equations, triangular fuzzy linear and nonlinear systems of equations also play a key role in determining the level of uncertainty. Convergence analysis illustrates that the proposed numerical technique's order of convergence for solving a triangular fuzzy system of linear and nonlinear equations is three. The newly developed numerical scheme was then applied to solve several triangular fuzzy boundary value problems. In terms of convergence rate, computing time, and residual error, numerical test problems indicate that the newly developed methods are more efficient than the current methods in the literature.

In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method’s global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature.

Quantum calculus can provide new insights into the nonlinear behaviour of functions and equations, addressing problems that may be difficult to tackle by classical calculus due to high nonlinearity. Iterative methods for solving nonlinear equations can benefit greatly from the mathematical theory and tools provided by quantum calculus, e.g., using the concept of q-derivatives, which extends beyond classical derivatives. In this paper, we develop parallel numerical root-finding algorithms that approximate all distinct roots of nonlinear equations by utilizing q-analogies of the function derivative. Furthermore, we utilize neural networks to accelerate the convergence rate by providing accurate initial guesses for our parallel schemes. The global convergence of the q-parallel numerical techniques is demonstrated using random initial approximations on selected biomedical applications, and the efficiency, stability, and consistency of the proposed hybrid numerical schemes are analyzed.

We proposed and analyzed the fractional simultaneous technique for approximating all the roots of nonlinear equations in this research study. The newly developed fractional Caputo-type simultaneous scheme’s order of convergence is 3ς+5, according to convergence analysis. Engineering-related numerical test problems are taken into consideration to demonstrate the efficiency and stability of fractional numerical schemes when compared to previously published numerical iterative methods. The newly developed fractional simultaneous approach converges on random starting guess values at random times, demonstrating its global convergence behavior. Although the newly developed method shows global convergent behavior when all starting guess values are distinct, the method diverges otherwise. The total computational time, number of iterations, error graphs and maximum residual error all clearly illustrate the stability and consistency of the developed scheme. The rate of convergence increases as the fractional parameter’s value rises from 0.1 to 1.0.

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

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.

Engineering and applied mathematics disciplines that involve differential equations include classical mechanics, thermodynamics, electrodynamics, and general relativity. Modelling a wide range of real-world situations sometimes comprises ambiguous, imprecise, or insufficient situational information, as well as multiindex, uncertainty, or restriction dynamics. As a result, intuitionistic fuzzy set models are significantly more useful and versatile in dealing with this type of data than fuzzy set models, triangular, or trapezoidal fuzzy set models. In this research, we looked at differential equations in a generalized intuitionistic fuzzy environment. We used the modified Adomian decomposition technique to solve generalized intuitionistic fuzzy initial value problems. The generalized modified Adomian decomposition technique is used to solve various higher-order generalized trapezoidal intuitionistic fuzzy initial value problems, circuit analysis problems, mass-spring systems, steam supply control sliding value problems, and some other problems in physical science. The outcomes of numerical test applications were compared to exact technique solutions, and it was shown that our generalized modified Adomian decomposition method is efficient, robotic, and reliable, as well as simple to implement.

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.

Finding all the roots of a nonlinear equation is an important and difficult task that arises naturally in numerous scientific and engineering applications. Sequential iterative algorithms frequently use a deflating strategy to compute all the roots of the nonlinear equation, as rounding errors have the potential to produce inaccurate results. On the other hand, simultaneous iterative parallel techniques require an accurate initial estimation of the roots to converge effectively. In this paper, we propose a new class of global neural network-based root-finding algorithms for locating real and complex polynomial roots, which exploits the ability of machine learning techniques to learn from data and make accurate predictions. The approximations computed by the neural network are used to initialize two efficient fractional Caputo-inverse simultaneous algorithms of convergence orders ς+2 and 2ς+4, respectively. The results of our numerical experiments on selected engineering applications show that the new inverse parallel fractional schemes have the potential to outperform other state-of-the-art nonlinear root-finding methods in terms of both accuracy and elapsed solution time.

Abstract In this article, we construct a family of iterative methods for finding a single root of nonlinear equation and then generalize this family of iterative methods for determining all roots of nonlinear equations simultaneously. Further we extend this family of root estimating methods for solving a system of nonlinear equations. Convergence analysis shows that the order of convergence is 3 in case of the single root finding method as well as for the system of nonlinear equations and is 5 for simultaneous determination of all distinct and multiple roots of a nonlinear equation. The computational cost, basin 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 literature.

Abstract Two new iterative methods for the simultaneous determination of all multiple as well as distinct roots of nonlinear polynomial equation are established, using two suitable corrections to achieve a very high computational efficiency as compared to the existing methods in the literature. Convergence analysis shows that the orders of convergence of the newly constructed simultaneous methods are 10 and 12. At the end, numerical test examples are given to check the efficiency and numerical performance of these simultaneous methods.

Abstract A highly efficient new three-step derivative-free family of numerical iterative schemes for estimating all roots of polynomial equations is presented. Convergence analysis proved that the proposed simultaneous iterative method possesses 12th-order convergence locally. Numerical examples and computational cost are given to demonstrate the capability of the method presented.

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.

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.

This paper is introducing a new concept of triangular linear Diophantine fuzzy numbers (TLDFNs) in a generic way. We first introduce the concept of TLDFNs and then study the arithmetic operations on these numbers. We find a method for the ranking of these TLDFNs. At the end, we formulate the linear and quadratic equations of the types A+X=B,A·X+B=C, and A·X2+B·X+C=D where the elements A,B,C, and D are TLDFNs. We provide a procedure for the solution of these equations using s,t,u,v-cut and also provide the examples.

In this research article, we propose efficient numerical iterative methods for estimating roots of interval-valued trapezoidal fuzzy nonlinear equations. Convergence analysis proves that the order of convergence of numerical schemes is 3. Some real-life applications are considered from optimization as numerical test problems which contain interval-valued trapezoidal fuzzy quantities in parametric form. Numerical illustrations are given to show the dominance efficiency of the newly constructed iterative schemes as compared to existing methods in literature.