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1.
Mudassir Shams Nasreen Kausar Shams Forruque Ahmed Irfan Anjum Badruddin Syed Javed 《计算机、材料和连续体（英文）》2023,74(3):5331-5347
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. 相似文献
2.
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. 相似文献
3.
Scott A. Burns Keith M. Mueller 《International journal for numerical methods in engineering》1999,46(12):1987-1996
A numerical method is presented for solving systems of non‐linear equations that contain some variables that are strictly positive and others that have no restriction on sign. Naturally positive variables arise frequently when modelling the behaviour of engineering systems, such as physical dimension, concentration of a chemical species, duration of an event, etc. When modelling systems of this type, it is also common to introduce additional variables that are not restricted in sign, such as stresses, displacements, velocities, accelerations, etc. Many numerical methods may experience performance difficulties due to the existence of spurious solutions which have negative components for one or more of the positive variables. Recently, the monomial method has been developed as an effective tool for systems with variables that are all strictly positive. This paper presents a hybrid method, combining the monomial method and Newton's method, for systems containing both types of variables. It is demonstrated that this hybrid method can be more effective in solving systems of equations with both positive and free variables than either method alone. Basins of attraction constructions are presented as a demonstration of the effectiveness of the hybrid method as applied to the design of a civil engineering frame structure. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
4.
5.
Ummu Atiqah Mohd Roslan Peter Ashwin 《Dynamical Systems: An International Journal》2016,31(3):375-392
We consider a piecewise expanding linear map with a Milnor attractor whose basin is riddled with the basin of a second attractor. To characterize the local geometry of this riddled basin, we calculate a stability index for points within the attractor as well as introducing a global stability index for the attractor as a set. Our results show that for Lebesgue almost all points in attractor, the index is positive and we characterize a parameter region, where some points have negative index. We show there exists a dense set of points for which the index is not converge. Comparing to recent results of Keller, we show that the stability index for points in the attractor can be expressed in terms of a global stability index for the attractor and Lyapunov exponents for this point. 相似文献
6.
Owe Axelsson Alexander Padiy 《International journal for numerical methods in engineering》1999,44(6):801-818
This paper presents a newly developed iterative algorithm for solving problems of linear isotropic elasticity discretized by means of mixed finite elements. It continues work started in References 1–5. The proposed method uses a pressure Schur complement approach to solve a saddle‐point system arising in the mixed formulation. As an inner solver for the displacement field variables it uses an extension to the robust black‐box multilevel procedure suggested in Reference 4. The proposed method works on a hierarchical sequence of finite element meshes to solve the problem with an arithmetic cost, nearly proportional to the dimension of the arising algebraic system. The coarsest mesh in the above sequence of meshes can consist of almost arbitrary triangular patches, which allows in practice to capture the solution even using a moderate number of successive refinement steps. The rate of convergence of the algorithm is bounded uniformly with respect to the problem coefficients, namely the Young's modulus E and the Poisson ratio ν. This makes it possible to apply the method for a broad class of engineering problems. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
7.
Steady Navier-Stokes equations are solved by three different space iterationmethods based on the lowest order nonconforming finite element pairs $mathscr{P}_1mathscr{N} mathscr{C}-mathscr{P}_1,$ including simple, Oseen, and Newton iterative methods. The stability and convergenceof these methods are studied, and their CPU time and numerical convergence rate arediscussed. Numerical results are in good agreement with theoretical findings. In particular, numerical experiments show that for large viscosity, the Newton method convergesfaster than to others, whereas the Oseen method is more suitable for the equations withsmall viscosity. 相似文献
8.
Michael J. Coleman 《Dynamical Systems: An International Journal》2010,25(2):215-238
We discuss the dynamics and stability of a rigid rimless spoked wheel, or regular polygon, confined to ‘rolling’ in a vertical plane uphill or downhill. The wheel has smooth inverted pendulum motions punctuated by repeated dissipative spoke impacts. It is a simple mechanical analogue to legged locomotion. The problem is completely soluble in closed form. We derive a return map for the full nonlinear system. The map has two asymptotically stable fixed points whose existence depends on slope angle: (1) standing still on two spokes and (2) limit cycle motion. For small slopes, only standing still exists; for intermediate slopes, the fixed points coexist; and, for large slopes, only limit cycles exist. We also completely define their basins of attraction. The rimless wheel's dynamical behaviour is analogous to two other dissipative systems, one smooth (a constantly forced, damped pendulum) and the other non-smooth (a 2D discrete skate). 相似文献
9.
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. 相似文献
10.
N. P. C. Marques J. C. F. Pereira 《International journal for numerical methods in engineering》2004,61(3):455-474
Six different preconditioning methods to accelerate the convergence rate of Krylov-subspace iterative methods are described, implemented and compared in the context of matrix-free techniques. The acceleration techniques comprehend Krylov-subspace iterative methods; invariant subspace-based methods and matrix approximations: Jacobi, LU-SGS, Deflated GMRES; Augmented GMRES; polynomial preconditioner and FGMRES/Krylov. The relative behaviour of the methods is explained in terms of the spectral properties of the resulting iterative matrix. The employed code uses a Newton–Krylov approach to iteratively solve the Euler or Navier–Stokes equations, for a supersonic ramp or a viscous compressible double-throat flow. The linear system approximate solver is the GMRES method, in either the restarted or FGMRES variants. The results show the better performance of the methods that approximate the iterative matrix, such as Jacobi, LU-SGS and FGMRES/Krylov. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
11.
Damiano Pasetto Massimiliano Ferronato Mario Putti 《International journal for numerical methods in engineering》2017,109(8):1159-1179
This paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive‐definite linear systems arising from the numerical discretization of transient parabolic and self‐adjoint partial differential equations. The preconditioners are obtained by nesting appropriate projections of reduced‐order models into the classical iteration of the preconditioned conjugate gradient (PCG). The main idea is to employ the reduced‐order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases. This approach is particularly appealing for transient systems where the full‐model solution has to be computed at each time step. In these cases, the natural reduced space is the one generated by full‐model solutions at previous time steps. When increasing the size of the projection space, the proposed methodology highly reduces the system conditioning number and the number of PCG iterations at every time step. The cost of the application of the preconditioner linearly increases with the size of the projection basis, and a trade‐off must be found to effectively reduce the PCG computational cost. The quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd. 相似文献
12.
Wasfi Shatanawi Ali Raza Muhammad Shoaib Arif Kamaledin Abodayeh Muhammad Rafiq Mairaj Bibi 《计算机、材料和连续体（英文）》2021,66(2):1121-1137
Nonlinear stochastic modeling plays a significant role in disciplines such as psychology, finance, physical sciences, engineering, econometrics, and biological sciences. Dynamical consistency, positivity, and boundedness are fundamental properties of stochastic modeling. A stochastic coronavirus model is studied with techniques of transition probabilities and parametric perturbation. Well-known explicit methods such as Euler Maruyama, stochastic Euler, and stochastic Runge–Kutta are investigated for the stochastic model. Regrettably, the above essential properties are not restored by existing methods. Hence, there is a need to construct essential properties preserving the computational method. The non-standard approach of finite difference is examined to maintain the above basic features of the stochastic model. The comparison of the results of deterministic and stochastic models is also presented. Our proposed efficient computational method well preserves the essential properties of the model. Comparison and convergence analyses of the method are presented. 相似文献
13.
Yuming Chu Naila Rafiq Mudassir Shams Saima Akram Nazir Ahmad Mir Humaira Kalsoom 《计算机、材料和连续体（英文）》2021,66(1):275-290
In this article, we construct the most powerful family of simultaneousiterative 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 freesimultaneous iterative method is nine. Our main aim is to check out the mostregularly used simultaneous iterative methods for finding all roots of non-linearequations by studying their dynamical planes, numerical experiments and CPUtime-methodology. Dynamical planes of iterative methods are drawn by usingMATLAB for the comparison of global convergence properties of simultaneousiterative methods. Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numericaltest examples. Numerical test examples, dynamical behavior and computationalefficiency are provided to present the performance and dominant efficiency ofthe newly constructed derivative free family of simultaneous iterative methodover existing higher order simultaneous methods in literature. 相似文献
14.
Zafar Iqbal Muhammad Aziz-ur Rehman Nauman Ahmed Ali Raza Muhammad Rafiq Ilyas Khan Kottakkaran Sooppy Nisar 《计算机、材料和连续体（英文）》2022,71(2):2141-2157
In this article, a brief biological structure and some basic properties of COVID-19 are described. A classical integer order model is modified and converted into a fractional order model with as order of the fractional derivative. Moreover, a valued structure preserving the numerical design, coined as Grunwald–Letnikov non-standard finite difference scheme, is developed for the fractional COVID-19 model. Taking into account the importance of the positivity and boundedness of the state variables, some productive results have been proved to ensure these essential features. Stability of the model at a corona free and a corona existing equilibrium points is investigated on the basis of Eigen values. The Routh–Hurwitz criterion is applied for the local stability analysis. An appropriate example with fitted and estimated set of parametric values is presented for the simulations. Graphical solutions are displayed for the chosen values of (fractional order of the derivatives). The role of quarantined policy is also determined gradually to highlight its significance and relevancy in controlling infectious diseases. In the end, outcomes of the study are presented. 相似文献
15.
Edmond Chow Thomas A. Manteuffel Charles Tong Bradley K. Wallin 《International journal for numerical methods in engineering》2003,57(8):1129-1144
Slide surface and contact boundary conditions can be implemented via Lagrange multipliers in the algebraic equations in implicit structural analysis. This indefinite set of equations is difficult to solve by iterative methods and is often too large to be solved by direct methods. When there are m constraints and there exists a set of m variables where each variable is only involved in a single constraint, we advocate a direct elimination technique which leaves a sparse, positive definite system to solve by iterative methods. We prove that the amount of ‘fill‐in’ created by this process is independent of the size of the slide surfaces. In addition, the eigenvalues of the reduced matrix do not differ significantly from the eigenvalues of the unconstrained matrix. This method can be extended to the case where constrained surfaces intersect and leads to a graph theoretic approach for determining which variables can be eliminated efficiently for constraints with more general structure. Published in 2003 by John Wiley & Sons, Ltd. 相似文献
16.
对数值微分问题构造了一类新的软化子求解方法,并在Ｌ＾２意义上证明了该方法可以提高软化解的收敛阶。 相似文献
17.
Vojin T. Jovanovic Kazem Kazerounian 《International journal for numerical methods in engineering》1998,42(4):729-747
This paper is an attempt to investigate the sensitivity to change of units and co-ordinate systems in computational kinematics when it involves both orientation and displacement in three dimensional space. The focus is on the behaviour of the Newton–Raphson iterative technique for solving customary system of equations for kinematic loop closure. It is shown that with the change of units or co-ordinate systems for some initial points the method does not converge to the same solution. Such behaviour is attributed to the shift of the boundaries of the so-called basins of attraction which play an important role in the theory of Chaos and Fractals. A number of numerical examples have been investigated and presented in tabulated form. To reduce the effect of sensitivity two procedures are suggested. One of them, Non-linear Elimination, is a recent development based on numerical elimination of variables in a system of equations. Investigation presented in this paper is the first of its kind and it is hoped that it will initiate further research to treat the problem of sensitivity in computational kinematics. © 1998 John Wiley & Sons, Ltd. 相似文献
18.
本文讨论了在无重根情况下,利用改进的Newton迭代法对一种同时求多项式零点的并行迭代法进行加速,得到了一种新的加速迭代法。首先证明了该方法是收敛的,并且理论证明出收敛阶至少是5阶;其次,分析了该方法的计算效率;最后通过实际的数值算例表明:计算收敛阶和定理结论是一致的,且本算法具有较高的计算效率。 相似文献
19.
Jürgen v. Hagen Werner Wiesbeck 《International journal for numerical methods in engineering》2002,54(3):317-329
An efficient method to solve electromagnetic scattering problems involving several metallic scatterers or bodies composed of dielectric and metallic regions is proposed. So far, the method of moments has successfully been applied to large arrays of identical scatterers when it was combined with preconditioned iterative algorithms to solve for the linear system of equations. Here, the method is generalized to geometries that are composed of several metallic elements of different shapes and sizes, and also to scatterers that are composed of metallic and dielectric regions. The method uses in its core an iterative algorithm, preferably the transpose‐free quasi‐minimum residual (TFQMR) algorithm, and a block diagonal Jacobi preconditioner. For best performance, the blocks for the preconditioner are chosen according to individual scatterers or groups of scatterers for the array case, and according to the electric and magnetic current basis functions for dielectric/metallic scatterers. The iterative procedure converges quickly for an optimally chosen preconditioner, and is robust even for a non‐optimal preconditioner. Reported run times are compared to run times of an efficiently programmed LU factorization, and are shown to be significantly lower. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
20.
关于一类插值多项式的最高收敛阶 总被引：3，自引：0，他引：3
以第一类Tchebyshev多项式的零点作为插值节点,推广了伯恩斯坦提出的一个问题,构造了插值多项式算子Gn,b(f;x),它不仅对f(x)∈C^a[-1,1](p≤a≤b-1,其中b为自然数）一致收敛,而且收剑阶达到了最佳。对算子Gn,b(f;x),最高收敛阶不会超过1/n^6,这是对伯恩斯坦所提出问题的一个圆满的回答。 相似文献