Efficient Numerical Scheme for Solving Large System of Nonlinear Equations

A fifth-order family of an iterative method for solving systems of nonlinear equations and highly nonlinear boundary value problems has been developed in this paper.Convergence analysis demonstrates that the local order of convergence of the numerical method is five.The computer algebra system CAS-Maple,Mathematica,or MATLAB was the primary tool for dealing with difficult problems since it allows for the handling and manipulation of complex mathematical equations and other mathematical objects.Several numerical examples are provided to demonstrate the properties of the proposed rapidly convergent algorithms.A dynamic evaluation of the presented methods is also presented utilizing basins of attraction to analyze their convergence behavior.Aside from visualizing iterative processes,this methodology provides useful information on iterations,such as the number of diverging-converging points and the average number of iterations as a function of initial points.Solving numerous highly nonlinear boundary value problems and large nonlinear systems of equations of higher dimensions demonstrate the performance,efficiency,precision,and applicability of a newly presented technique.

Ostrowski’s Method for Solving Nonlinear Equations and Systems

The dynamic characteristics and the efficiency of the Ostrowski’s method allow it to be crowned as an excellent tool for solving nonlinear problems.This article shows different versions of the classic method that allow it to be applied to a wide range of engineering problems.Among them stands out the derivative-free definition applying divided differences,the introduction of memory and its extension to the resolution of nonlinear systems of equations.All of these versions are compared in a numerical simulations section where the results obtained are compared with other classic methods.

SOLVERS FOR SYSTEMS OF LARGE SPARSE LINEAR AND NONLINEAR EQUATIONS BASED ON MULTI-GPUS

Numerical treatment of engineering application problems often eventually results in a solution of systems of linear or nonlinear equations.The solution process using digital computational devices usually takes tremendous time due to the extremely large size encountered in most real-world engineering applications.So,practical solvers for systems of linear and nonlinear equations based on multi graphic process units（GPUs）are proposed in order to accelerate the solving process.In the linear and nonlinear solvers,the preconditioned bi-conjugate gradient stable（PBi-CGstab）method and the Inexact Newton method are used to achieve the fast and stable convergence behavior.Multi-GPUs are utilized to obtain more data storage that large size problems need.

Improved Nonlinear Equation Method for Numerical Prediction of Jominy End-Quench Curves

Without considering the effects of alloying interaction on the Jominy end-quench curves, the prediction resuits obtained by YU Bai-hai＇s nonlinear equation method for multi-alloying steels were different from those experimental ones reported in literature. Some alloying elements have marked influence on Jominy end-quench curves of steels. An improved mathematical model for simulating the Jominy end-quench curves is proposed by introducing a parameter named alloying interactions equivalent （Le）. With the improved model, the Jominy end-quench curves of steels so obtained agree very well with the experimental ones.

A Family of Fifth-order Iterative Methods for Solving Nonlinear Equations

In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects.

New Optimal Newton-Householder Methods for Solving Nonlinear Equations and Their Dynamics

The classical iterative methods for finding roots of nonlinear equations,like the Newton method,Halley method,and Chebyshev method,have been modified previously to achieve optimal convergence order.However,the Householder method has so far not been modified to become optimal.In this study,we shall develop two new optimal Newton-Householder methods without memory.The key idea in the development of the new methods is the avoidance of the need to evaluate the second derivative.The methods fulfill the Kung-Traub conjecture by achieving optimal convergence order four with three functional evaluations and order eight with four functional evaluations.The efficiency indices of the methods show that methods perform better than the classical Householder’s method.With the aid of convergence analysis and numerical analysis,the efficiency of the schemes formulated in this paper has been demonstrated.The dynamical analysis exhibits the stability of the schemes in solving nonlinear equations.Some comparisons with other optimal methods have been conducted to verify the effectiveness,convergence speed,and capability of the suggested methods.

A Nonmonotone Trust Region Method for Solving Symmetric Nonlinear Equations

A trust region method combining with nonmonotone technique is proposed tor solving symmetric nonlinear equations. The global convergence of the given method will be established under suitable conditions. Numerical results show that the method is interesting for the given problems.

Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations

There are several ways that can be used to classify or compare iterative methods for nonlinear equations,for instance;order of convergence,informational efficiency,and efficiency index.In this work,we use another way,namely the basins of attraction of the method.The purpose of this study is to compare several iterative schemes for nonlinear equations.All the selected schemes are of the third-order of convergence and most of them have the same efficiency index.The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees.As a comparison,we determine the CPU time(in seconds)needed by each scheme to obtain the basins of attraction,besides,we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all methods.Comparisons confirm the fact that basins of attraction differ for iterative methods of different orders,furthermore,they vary for iterative methods of the same order even if they have the same efficiency index.Consequently,this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index.

A Study for Obtaining New and More General Solutions of Special-Type Nonlinear Equation

The generalized algebraic method with symbolic computation is extended to some special-type nonlinear equations for constructing a series of new and more general travelling wave solutions in terms of special functions. Such equations cannot be directly dealt with by the method and require some kinds of pre-processing techniques. It is shown that soliton solutions and triangular periodic solutions can be established as the limits of the Jacobi doubly periodic wave solutions.

Integrating Variable Reduction Strategy With Evolutionary Algorithms for Solving Nonlinear Equations Systems

Nonlinear equations systems(NESs)are widely used in real-world problems and they are difficult to solve due to their nonlinearity and multiple roots.Evolutionary algorithms(EAs)are one of the methods for solving NESs,given their global search capabilities and ability to locate multiple roots of a NES simultaneously within one run.Currently,the majority of research on using EAs to solve NESs focuses on transformation techniques and improving the performance of the used EAs.By contrast,problem domain knowledge of NESs is investigated in this study,where we propose the incorporation of a variable reduction strategy(VRS)into EAs to solve NESs.The VRS makes full use of the systems of expressing a NES and uses some variables(i.e.,core variable)to represent other variables(i.e.,reduced variables)through variable relationships that exist in the equation systems.It enables the reduction of partial variables and equations and shrinks the decision space,thereby reducing the complexity of the problem and improving the search efficiency of the EAs.To test the effectiveness of VRS in dealing with NESs,this paper mainly integrates the VRS into two existing state-of-the-art EA methods(i.e.,MONES and DR-JADE)according to the integration framework of the VRS and EA,respectively.Experimental results show that,with the assistance of the VRS,the EA methods can produce better results than the original methods and other compared methods.Furthermore,extensive experiments regarding the influence of different reduction schemes and EAs substantiate that a better EA for solving a NES with more reduced variables tends to provide better performance.

On Newton-Like Methods for Solving Nonlinear Equations

In this paper, we present a family of general New to n-like methods with a parametric function for finding a zero of a univariate fu nction, permitting f′(x)=0 in some points. The case of multiple roots is n ot treated. The methods are proved to be quadratically convergent provided the w eak condition. Thus the methods remove the severe condition f′(x)≠0. Based on the general form of the Newton-like methods, a family of new iterative meth ods with a variable parameter are developed.

Some geometrical iteration methods for nonlinear equations

This paper describes geometrical essentials of some iteration methods （e.g. Newton iteration, secant line method, etc.） for solving nonlinear equations and advances some geometrical methods of iteration that are flexible and efficient.

THE MAPPING RELATION AMONG SOLUTION OF SOME NONLINEAR EQUATIONS

Among the solutions of three kinds of nonlinear equations in one dimensional systems, cubic nonlinear Klein-Gordon (including Φ~4), Sine-Gordon and double Sine-Gordon, some mapping relations exist. When a solution of any one equation is known, so are the other two.

Construction of solitonary and periodic solutions to some nonlinear equations using EXP-function method

This paper applies the EXP-function method to find exact solutions of various nonlinear equations. Tzitzeica- Dodd-Bullough (TDB) equation was selected to illustrate the effectiveness and convenience of the suggested method. More generalized solitonary solutions with free parameters were obtained by suitable choice of the free parameters, and also the obtained solitonary solutions can be converted into periodic solutions.

ANALYTICAL FORMULAS OF SOLUTIONS OF GEOMETRICALLY NONLINEAR EQUATIONS OF AXISYMMETRIC PLATES AND SHALLOW SHELLS

Starting from the step-by-step iterative method, the analytical formulas of solutions of the geometrically nonlinear equations of the axisymmetric plates and shallow shells, have been obtained. The uniform convergence of the iterative method, is used to prove the convergence of the analytical formulas of the exact solutions of the equa- tions.

A weak condition for secant method to solve systems of nonlinear equations

In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.

Different-Periodic Travelling Wave Solutions for Nonlinear Equations

Using Jacobi elliptic function linear superposition approach for the (1+1)-dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation and the (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equation, many new periodic travelling wave solutions with different periods and velocities are obtained based on the known periodic solutions. This procedure is crucially dependent on a sequence of cyclic identities involving Jacobi elliptic functions sn(), cn(), and dn().

HBFTrans2: A Maple Package to Construct Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear form of nonlinear equations by a logarithm transformation is presented. The improved algorithm is more efficient by using the property of Hirota-D operator. The software package HBFTrans2 is written in Maple and its running efficiency is tested by a variety of soliton equations.

Design Principles-Based Interactive Learning Tool for Solving Nonlinear Equations

Interactive learning tools can facilitate the learning process and increase student engagement,especially tools such as computer programs that are designed for human-computer interaction.Thus,this paper aims to help students learn five different methods for solving nonlinear equations using an interactive learning tool designed with common principles such as feedback,visibility,affordance,consistency,and constraints.It also compares these methods by the number of iterations and time required to display the result.This study helps students learn these methods using interactive learning tools instead of relying on traditional teaching methods.The tool is implemented using the MATLAB app and is evaluated through usability testing with two groups of users that are categorized by their level of experience with root-finding.Users with no knowledge in root-finding confirmed that they understood the root-finding concept when interacting with the designed tool.The positive results of the user evaluation showed that the tool can be recommended to other users.

A Maple Package on Symbolic Computation of Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations withlogarithmic transformations is presented.In the algorithm,the general assumption of Hirota bilinear form is successfullyreduced based on the property of uniformity in rank.Furthermore,we discard the integral operation in the traditionalalgorithm.The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety solitonequations.