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.     

Computer Methodologies for the Comparison of Some Efficient Derivative Free Simultaneous Iterative Methods for Finding Roots of Non-Linear Equations

In this article, we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations. Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine. Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes, numerical experiments and CPU time-methodology. Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods. Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples. Numerical test examples, dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.     

An Iterative Scheme of Arbitrary Odd Order and Its Basins of Attraction for Nonlinear Systems

In this paper, we propose a fifth-order scheme for solving systems of nonlinear equations. The convergence analysis of the proposed technique is discussed. The proposed method is generalized and extended to be of any odd order of the form 2n − 1. The scheme is composed of three steps, of which the first two steps are based on the two-step Homeier’s method with cubic convergence, and the last is a Newton step with an appropriate approximation for the derivative. Every iteration of the presented method requires the evaluation of two functions, two Fréchet derivatives, and three matrix inversions. A comparison between the efficiency index and the computational efficiency index of the presented scheme with existing methods is performed. The basins of attraction of the proposed scheme illustrated and compared to other schemes of the same order. Different test problems including large systems of equations are considered to compare the performance of the proposed method according to other methods of the same order. As an application, we apply the new scheme to some real-life problems, including the mixed Hammerstein integral equation and Burgers’ equation. Comparisons and examples show that the presented method is efficient and comparable to the existing techniques of the same order.     

A New Optimal Iterative Algorithm for Solving Nonlinear Poisson Problems in Heat Diffusion

The nonlinear Poisson problems in heat diffusion governed by elliptic type partial differential equations are solved by a modified globally optimal iterative algorithm (MGOIA). The MGOIA is a purely iterative method for searching the solution vector x without using the invert of the Jacobian matrix D. Moreover, we reveal the weighting parameter αc in the best descent vector w = αcE + DTE and derive the convergence rate and find a criterion of the parameter γ. When utilizing αc and γ, we can further accelerate the convergence speed several times. Several numerical experiments are carefully discussed and validated the proposed method.     

非线性方程组的牛顿-整体松弛并行多分裂法

松弛技术是提高分裂迭代法收敛速度的一种基本技术。本文在前人工作的基础上,把求解线性方程组的松弛型矩阵多分裂迭代法推广到了求解非线性方程组,并通过引入多个松弛因子,提出了整体松弛的概念和方法。进而,文中研究了牛顿—整体松弛型矩阵多分裂TOR迭代法,建立了其局部收敛性定理,给出了收敛速度的估计。对于本文提出的求解非线性方程组的牛顿—整体松弛型多分裂TOR迭代法,当选取近似最优参数时,我们的方法将比其他方法有更快的收敛速度。     

一类非线性代数方程组的Newton-Triangle Splitting迭代法

Triangle Splitting迭代方法是求解大型稀疏非Hermitian正定线性代数方程组的一种有效迭代算法.为了有效求解大型稀疏且Jacobi矩阵为非Hermitian正定的非线性代数方程组,本文将Triangle Splitting迭代方法作为不精确Newton方法的内迭代求解器,构造了不精确Newton-Triangle Splitting迭代方法.在适当的约束条件下,给出了该方法的两类局部收敛性定理.通过数值实验结果验证了该方法的可行性和有效性,并说明了该方法在计算时间和迭代次数方面比Newton-BTSS迭代方法更有优势.     

Banach空间非线性混合型微分-积分方程解的存在唯一性

在较宽的条件下研究了Banach空间中非线性混合型微分—积分初值问题解的存在唯一性及解的迭代逼近和误差估计，改进并推广了最近的一些结果。     

Dynamical Newton-Like Methods with Adaptive Stepsize for Solving Nonlinear Algebraic Equations

In this paper, a dynamical Newton-like method with the adaptive stepsize based on the construction of a scalar homotopy function to transform a vector function of non-linear algebraic equations (NAEs) into a time-dependent scalar function by introducing a fictitious time-like variable is proposed. With the introduction of the fictitious time-like function, we derived the adaptive stepsize using the dynamics of the residual vector. Based on the proposed dynamical Newton-like method, we can also derive the dynamical Newton method (DNM) and the dynamical Jacobian-inverse free method (DJIFM) with the transformation matrix as the inverse of the Jacobian and the identity matrix, respectively. These two dynamical Newton-like methods are then adopted for the solution of NAEs. Numerical illustrations demonstrate that taking advantages of the dynamical Newton-like method with the adaptive stepsize the proposed two dynamical Newton-like methods can release limitations of the conventional Newton method such as root jumping, the divergence at inflection points, root oscillations, and the divergence of the root. Results reveal that with the use of the fictitious time-like function the proposed method presents exponential convergence. In addition, taking the advantages of the transformation matrix, the proposed method does not need to calculate the inverse of the Jacobian matrix and thus has great numerical stability.     

一类二阶非线性差分方程解的振动性

研究了一类比较广泛的二阶非线性差分方程，得出了其解振动的平均型判定准则，所得结果包含和推广了已有的结果。     

Non-smooth Nonlinear Equations Methods for Solving 3D Elastoplastic Frictional Contact Problems

Based on the non-smooth nonlinear equations method for modeling three-dimensional elastic frictional contact problems (hereafter called NNEM), the extension to elastoplastic case in which the material nonlinearity is also involved is presented in this paper. Two approaches which combine two methods for solving elastoplastic problem with NNEM are proposed. A Numerical example is given to demonstrate the validation of the approaches.     

延迟积分微分方程单支方法的稳定性分析

本文研究求解非线性延迟积分微分方程的单支方法的数值稳定性,其中积分部分采用复化梯形公式计算。分析表明:在一定条件下,A-稳定的单支方法是数值稳定的,而强A-稳定的单支方法是渐近稳定的。最后,数值试验验证了本文所获理论结果的正确性。     

高阶非线性中立型差分方程组多正解的存在性

本文研究了一类高阶非线性中立型差分方程组多正解的存在性。通过构造实Banach空间中的严格集压缩算子及利用不动点指数理论,得到了这类方程组两个正解的存在性准则。所得结论推广并改进了已有的相关结果。     

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 performbetter 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 Class of Preconditioned TGHSS-Based Iteration Methods for Weakly Nonlinear Systems

In this paper, we first construct a preconditioned two-parameter generalized Hermitian and skew-Hermitian splitting (PTGHSS) iteration method based on the two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method for non-Hermitian positive definite linear systems. Then a class of PTGHSS-based iteration methods are proposed for solving weakly nonlinear systems based on separable property of the linear and nonlinear terms. The conditions for guaranteeing the local convergence are studied and the quasi-optimal iterative parameters are derived. Numerical experiments are implemented to show that the new methods are feasible and effective for large scale systems of weakly nonlinear systems.     

一类非线性多时滞脉冲抛物型方程解的振动性质

本文研究一类非线性多时滞脉冲抛物型方程在齐次Dirichlet和Neumann边界条件下解的振动性质.利用分析技巧,给出一个脉冲微分不等式无最终正解(或最终负解)的条件.然后,利用平均法,将该方程解振动性问题转化为相应脉冲时滞微分不等式有无最终正解(或最终负解)问题,进而在两类齐次边界条件下获得了判别该类方程解振动的充...     

具有偏差变元的二阶非线性微分方程的无界解(英文)

给出了关于具有偏差变元的二阶非线性微分方程无界非振动解存在性的充分条件     

一类具有时滞的非线性微分代数系统的谱方法

首先给出了一类带有时滞的非线性微分代数系统(DDAE)存在惟一解的充分条件:然后将原问题变形为无时滞的带有非局部边值条件的偏微分系统;接下来运用Chebyshev谱方法求得原问题的近似解。所用方法的优点在于避免了许多在处理时滞项时所遇到的数值困难,而且谱方法具有高精度;最后,给出了实例来说明所提出的方法的可行性。     

Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions

This article is devoted to the study of some high-order difference schemes for the distributed-order time-fractional equations in both one and two space dimensions. Based on the composite Simpson formula and Lubich second-order operator, a difference scheme is constructed with $\mathscr{O}(τ^2+h^4+σ^4)$ convergence in the $L_1$($L_∞$)-norm for the one-dimensional case, where $τ$, $h$ and $σ$ are the respective step sizes in time, space and distributed-order. Unconditional stability and convergence are proven. An ADI difference scheme is also derived for the two-dimensional case, and proven to be unconditionally stable and $\mathscr{O}(τ^2|lnτ|+h^4_1+h^4_2+σ^4)$ convergent in the $L_1$($L_∞$)-norm, where $h_1$ and $h_2$ are the spatial step sizes. Some numerical examples are also given to demonstrate our theoretical results.     

Efficient and Stable Numerical Methods for Multi-Term Time Fractional Sub-Diffusion Equations

Some efficient numerical schemes are proposed for solving one-dimensional (1D) and two-dimensional (2D) multi-term time fractional sub-diffusion equations, combining the compact difference approach for the spatial discretisation and $L1$ approximation for the multi-term time Caputo fractional derivatives. The stability and convergence of these difference schemes are theoretically established. Several numerical examples are implemented, testifying to their efficiency and confirming their convergence order.     

半正定线性系统广义非定常多分裂二阶段迭代方法的收敛性

为了高效求解正定或半正定的大型稀疏线性方程组,在第一阶段采用经典矩阵分裂的基础上,广义非定常多分裂二阶段迭代方法的第二阶段分裂融合了多分裂和矩阵预处理技术,对非定常多分裂二阶段迭代方法进行了推广。为了研究收敛性,将该迭代方法的算法形式和逻辑语言表达形式改写为紧凑的迭代格式。由此得到,广义非定常多分裂二阶段迭代算法在一个充分条件下收敛。最后,具有五对角系数矩阵的大型稀疏线性系统的数值算例验证了广义非定常多分裂二阶段迭代算法的普适性,并且从迭代次数和\,CPU\,时间上体现了算法的高效性。