Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung–Traub’s conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basins of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basins of attraction.

In this paper, a family of Newton-type iterative methods with memory is obtained for solving nonlinear equations, which uses some special self-accelerating parameters. To this end, we first present two optimal fourth-order iterative methods with memory for solving nonlinear equations. Then we give a novel way to construct the self-accelerating parameter and obtain a family of Newton-type iterative methods with memory. The self-accelerating parameters have the properties of simple structure and easy calculation, which do not increase the computational cost of the iterative methods. The convergence order of the new iterative method is increased from 4 to . Numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the new methods. Experiment results show that, compared with the existing methods, the new iterative methods with memory have the advantage of costing less computing time.

Based on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-order algorithms for solving one-variable equations. The new methods are fourth-, eighth-, and sixteenth-order converging and require at each iteration three, four, and five function evaluations, respectively. Therefore, all these algorithms are optimal in the sense of Kung–Traub conjecture; the new schemes have an efficiency index of 1.587, 1.682, and 1.741, respectively. We have given convergence analyses of the proposed methods and also given comparisons with already established known schemes having the same convergence order, demonstrating the efficiency of the present techniques numerically. We also studied basins of attraction to demonstrate their dynamical behavior in the complex plane.

The prime objective of this paper is to design a new family of optimal eighth-order iterative methods by accelerating the order of convergence of the existing seventh-order method without using more evaluations for finding simple root of nonlinear equations. Numerical comparisons have been carried out to demonstrate the efficiency and performance of the proposed method. Finally, we have compared new method with some existing eighth-order methods by basins of attraction and observed that the proposed scheme is more efficient.

Physics and applied mathematics form the basis for understanding natural phenomena using differential equations depicting the flow in porous media, the motion of viscous liquids, and the propagation of waves. These equations provide a thorough study of physical processes, enhancing the understanding of complex applications in engineering, technology, and medicine. This paper presents novel approximate solutions for the Darcy-Brinkmann-Forchheimer moment equation, the Blasius equation and the Falkner-Skan equation with initial / boundary conditions by using two iterative methods: the variational iteration method and the optimal variational iteration method. The variational iteration method is effectively developed by adding a control parameter to enhance the convergence speed and prevent large-scale divergence. The influence of physical parameters on the accuracy of the solution was also analyzed, since it was noted that increasing some parameters improves accuracy, while increasing others leads to a decrease the accuracy. Also, the convergence of the proposed methods has been discussed and proved. Moreover, comparison was made with some approximate methods available in the literature were used the operational matrices methods include: Bernstein’s method (BOM), Bernoulli’s method (BrOM), and the shifted Legendre’s method (LOM). Furthermore, the maximum values of the residual error were computed for the proposed methods and others operational matrices methods for different cases. The results demonstrated the efficiency and accuracy of the optimal variational iteration method in solving nonlinear ordinary differential equations in comparison to other methods. All calculations in this paper were made using the Mathematica®14 software.

Iterative methods are crucial in numerical analysis for approximating solutions to nonlinear problems encountered in various fields, such as biomathematics, thermodynamics, chemical engineering, and fluid mechanics. This paper introduces innovative iterative methods of convergence orders three and six to eight, developed using the homotopy perturbation technique (HPT). We address the limitations of traditional derivative-based methods by incorporating divided differences, resulting in derivative-free schemes with optimal convergence properties. We present four newly designed algorithms (HPM1, HPM2, HPM3, and HPM4) and provide a comprehensive convergence analysis. Numerical simulations and basins of attraction demonstrate the superior performance of the proposed methods compared to existing methods of the same order. The proposed methods exhibit faster convergence and larger basins of attraction, making them highly effective for solving nonlinear equations. Our new numerical methods rely on approximations and iterative processes, with each small step bringing us closer to the exact solution. Exploring these numerical methods encourages us to apply mathematics to solve challenging problems in physics, engineering, finance, and more.

In this paper, a new parametric class of optimal fourth-order iterative methods to estimate the solutions of nonlinear equations is presented. After the convergence analysis, a study of the stability of this class is made using the tools of complex discrete dynamics, allowing those elements of the class with lower dependence on initial estimations to be selected in order to find a very stable subfamily. Numerical tests indicate that the stable members perform better on quadratic polynomials than the unstable ones when applied to other non-polynomial functions. Moreover, the performance of the best elements of the family are compared with known methods, showing robust and stable behaviour.

In this paper, based on the stabilization technique, the Oseen iterative method and the two-level finite element algorithm are combined to numerically solve the stationary incompressible magnetohydrodynamic (MHD) equations. For the low regularity of the magnetic field, when dealing with the magnetic field sub-problem, the Lagrange multiplier technique is used. The stabilized method is applied to approximate the flow field sub-problem to circumvent the inf-sup condition restrictions. One- and two-level stabilized finite element algorithms are presented, and their stability and convergence analysis is given. The two-level method uses the Oseen iteration to solve the nonlinear MHD equations on a coarse grid of size H, and then employs the linearized correction on a fine grid with grid size h. The error analysis shows that when the grid sizes satisfy , the two-level stabilization method has the same convergence order as the one-level one. However, the former saves more computational cost than the latter one. Finally, through some numerical experiments, it has been verified that our proposed method is effective. The two-level stabilized method takes less than half the time of the one-level one when using the second class Nédélec element to approximate magnetic field, and even takes almost a third of the computing time of the one-level one when adopting the first class Nédélec element.

The present article deals with the effect of convexity in the study of the well-known Whittaker iterative method, because an iterative method converges to a unique solution \(t^*\) of the nonlinear equation \(\psi(t)=0\) faster when the function's convexity is smaller. Indeed, fractional iterative methods are a simple way to learn more about the dynamic properties of iterative methods, i.e., for an initial guess, the sequence generated by the iterative method converges to a fixed point or diverges. Often, for a complex root search of nonlinear equations, the selective real initial guess fails to converge, which can be overcome by the fractional iterative methods. So, we have studied a Caputo fractional double convex acceleration Whittaker's method (CFDCAWM) of order at least (\(1+2\zeta\)) and its global convergence in broad ways. Also, the faster convergent CFDCAWM method provides better results than the existing Caputo fractional Newton method (CFNM), which has (\(1+\zeta\)) order of convergence. Moreover, we have applied both fractional methods to solve the nonlinear equations that arise from different real-life problems.

A survey on the existing techniques used to design optimal iterative schemes for solving nonlinear equations is presented. The attention is focused on such procedures that use some evaluations of the derivative of the nonlinear function. After introducing some elementary concepts, the methods are classified depending on the optimal order reached and also some general families of arbitrary order are presented. Later on, some techniques of complex dynamics are introduced, as this is a resource recently used for many authors in order to classify and compare iterative methods of the same order of convergence. Finally, some numerical test are made to show the performance of several mentioned procedures and some conclusions are stated.

A new free derivative iterative method is presented in this article. The method is developed by combining Newton’s method and Euler’s method. Deriva- tives in this method are approximated by forward difference, hyperbola and divided difference. The order of convergence is proven analytically to be of sixth order. Numerical results exhibit that the new method is comparable to other methods. Basins of attraction are also provided to support the proposed method.

There are several measures for comparing methods for solving a single nonlinear equation. The first is the order of convergence, then the cost to achieve such rate. This cost is measured by counting the number of functions (and derivatives) evaluated at each step. After that, efficiency is defined as a function of the order of convergence and cost. Lately, the idea of basin of attraction is used. This shows how far one can start and still converge to the root. It also shows the symmetry/asymmetry of the method. It was shown that even methods that show symmetry when solving polynomial equations are not so when solving nonpolynomial ones. We will see here that the Euler–Cauchy method (a member of the Laguerre family of methods for multiple roots) is best in the sense that the boundaries of the basins have no lobes. The symmetry in solving a polynomial equation having two roots at ±1 with any multiplicity is obvious. In fact, the Euler–Cauchy method converges very fast in this case. We compare one member of a family of fifth-order methods for multiple roots with several well-known lower-order and efficient methods. We will show using a basin of attraction that the fifth-order method cannot compete with most of those lower-order methods.

Recently, some optimal fourth-order iterative methods for multiple roots of nonlinear equations are presented when the multiplicity m of the root is known. Different from these optimal iterative methods known already, this paper presents a new family of iterative methods using the modified Newton’s method as its first step. The new family, requiring one evaluation of the function and two evaluations of its first derivative, is of optimal order. Numerical examples are given to suggest that the new family can be competitive with other fourth-order methods and the modified Newton’s method for multiple roots.

Improved Chebyshev–Halley methods with sixth and eighth order convergence

This article concerned with the issue of solving a nonlinear equation with the help of iterative method where no any derivative evaluation is required per iteration.Therefore, this work contributes to a new class of optimal eighth-order Steffensen-type methods.Theoretical proof has been given to reveal the eighth-order convergence.Numerical comparisons have been carried out to show the effectiveness of contributed scheme.