Search

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. [ABSTRACT FROM AUTHOR]

This study extended an existing semi-analytical technique, the Homotopy Perturbation Method, to the Block Homotopy Modified Perturbation Method by solving two $ n \times n $ n × n 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 $ n \times n $ n × n TIF linear systems as it only requires the non-singularity of the $ n \times n $ n × n 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Solving fractional-order nonlinear equations is crucial in engineering, where precision and accuracy are essential. This study introduces a novel fractional parallel technique for solving nonlinear equations. To enhance convergence, we incorporate a simple root-finding method of order 3γ + 1 as a correction term in the parallel scheme. Theoretical analysis shows that the parallel scheme achieves a convergence order of 6γ + 3. Using a dynamical system approach, we identify optimal parameter values, and the symmetry in the dynamical planes for different fractional parameters demonstrates the method's stability and consistency in handling nonlinear problems. These parameter values are applied to the parallel scheme, yielding highly consistent results. Several engineering problems are examined to assess the method's efficiency, stability, and consistency compared to existing methods. [ABSTRACT FROM AUTHOR]

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 2 ψ + 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 2 ς + 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 FROM AUTHOR]

Fractional calculus can be used to fully describe numerous realworld 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 ϑ + 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. [ABSTRACT FROM AUTHOR]